Dave,
A few suggestions and long-winded comments:
1) Ideally, your sump pump is tied into underground tile from a perimeter drain system. When you get the water level in the sump very low, then you can do some investigating to determine if this is true. You should be able to see the circular cutouts in the plastic sump basin where the tile connect to it. In a corner installation, you will typically have 2 tie-ins with one tile entering parallel to each of the two walls forming the corner.
2) If you anticipate heavy spring rains and/or you already have a problem, then you want to force the pump to evacuate as much water as possible from the sump. Set the float or override the float if necessary. If you override the float, then you need to be in the basement nearby to make certain that you don't run the pump dry. You can override the float very easily - just get a piece of string and tie up the float so that it can't lower (temporarily!).
3) The gravel bed under your concrete basement floor functions as a buffer zone for water. When you evacuate as much water as possible from this area and from the soil under your house, then you have a cushion for the next storm. Generally, ground water must fill these buffer areas before the water can begin to infiltrate your basement. And with good perimeter drain tile and a properly functioning sump pump system, you will usually be able to remove water from these buffer areas fast enough to prevent them from overfilling during and after a heavy rain. Think of the area under your basement floor as a huge drywell. Note that this buffering function is dependent upon how well your sump pump is evacuating water under the entire basement floor and the surrounding soil. This will depend upon the depth of the gravel under your basement, the existence and current quality of the perimeter drain system, soil type, etc.
The more you lower the water level in these buffer areas, the more you are paying in electricity for the extra cushion of safety that you achieve. Personally, I feel that having the sump pump run more and avoiding a basement flood is well worth the extra bit of cost. We all pay for auto insurance, life insurance and home insurance. Setting your sump float to the lowest possible setting is just one more form of insurance.
4) You obviously have had a sump pump & float system which wasn't functioning optimally. You also may have no drain tile feeding the sump or a tile system which is clogged. If you have drain tile and you continue to have problems, then a "Roto Rooter" type of company may be able to come in and examine your tile system with a miniature camera. They may also be able to clear some blockages in the drain tile system, or locate areas of collapsed drain tile and address them (jackhammer, etc.)
5) For now, you should set you sump pump to the lowest reasonable water-level setting and see what happens. When the sump pit water level is low, examine the pit to see if you have drain tile coming into the pit. You can also hook up your old sump pump to function as a second pump if there is enough room in the pit. Buy some cheap flexible corrugated black plastic pipe so that the old pump can pump water to a safe location outdoors via a basement window, or have it drain into your basement drain or utility sink if you have city hookup and not a septic system for your sewer drains. Remove the float from the old pump and lock the remaining float arm in the "always running" mode. Using the old sump pump will double your pumping capability when you need it. This will provide you with valuable information about how well you can alleviate you water problems via the single sump pit that you have.
6) If you continue to have problems, then you must: a) Clear blocked drain tile if that is a problem as discussed above. b) Address all outside issues if possible. I have a very nice neighbor who is uphill from me. He had one roof drain which was dumping directly on the ground and the water was running onto my property & eventually to my basement. He fixed the problem by rerouting the water and helped me considerably. As you have guessed in your situation, it is possible for your water being dumped 6' from your house to still contribute to your problem. Then again, maybe not. The water entering your basement can be coming from far away, but it can also be coming from roof water that you are dumping 6' from your foundation. The biggest problem about addressing basement water issues is that we are forced to intelligently guess about what is happening out-of-sight and underground. c) You may need to install perimeter drains to dump into the sump pit. This is expensive. You can cut cost by hiring moonlighters from a basement company who will work off the books on a weekend. You can do it yourself if you are ambitious. The cost of materials is rather modest - this is labor intensive. Professionals will want to do a "complete system", whereas you can experiment with partial solutions. Adding drain tile along one or both of the two walls that would "feed" you sump pump may help tremendously. You can add drain tile in stages along additional walls as needed. d) You may need a second sump pit and pump on the opposite side of the basement. As above, this is a bit expensive to hire out, but can be done rather cheaply. I intend to install 2 sump pumps & pits this summer with help from my son. The cost for 2 will be just a few hundred dollars since I already have perimeter drain tile but no sump pumps. e) Use caution when taking the advise of "professionals." Obviously, it is difficult for those who sell basement waterproofing systems to be objective when advising you. This is an industry which makes a living through scare tactics and oversell. It is extremely common for homeowners to pay $10,000 or more when the actual fix that was needed could have been done for less than $1000. Also, remember that a 5 year, 10 year or lifetime warranty may be totally worthless if the company that did the work goes bankrupt, changes hands, restructures, etc. Even if you don't anticipate hiring such companies, it is extremely worthwhile to observe how they operate when installing systems for neighbors, friends and relatives. I have learned considerably by observing in many such cases. The most recent example was my next-door neighbor who probably paid between $8,000 and $12,000 for a "total system" whereas he and I could have addressed 95% of his problem with just one half day's labor and $25 in materials. From my observations, the folks who are "professionals" form companys with extremely poorly paid "grunts" who do the heavy manual labor and who are very poor at evaluating situations that they encounter. Project supervisors aren't much more astute. Even the sales reps and company bosses are rather obtuse in evaluating situations. They are experts at following a scipt which involves over-engineering every situation. Think of the car mechanic who will guarantee his work when fixing an auto which has an engine which isn't performing optimally. The mechanic is an expert at replacing spark plug wires, spark plugs, air filters, pcv valves, etc. He is also an expert at remove fuel injection systems or carb systems and sending them our for rebuilds. Etc., etc. Of course, when he replaces every replacable component in the fuel delivery and ignition systems, the car will probably run much better than before. But you may be paying $1200 for a problem which could have been fixed for $50. I have seen auto shops which specialize in such extreme approaches to every situation in which an auto runs poorly. Sadly, the approach of almost ever "pro" toward basement water problems is very similar.
Good luck, Gideon