It would be very difficult to make a useful comment without seeing the contract that you've signed. These are usually boilerplate documents, many fuel companies have them on their sites, perhaps you can point us to the one that you're using if it's available online.
But in general terms, I've looked at my own (LPG though, not oil) and several other automatic oil delivery contracts I could access online, and they are normally specifically denying responsibility for freeze-ups under any circumstances, including running out of fuel. In other words, if you did have pipe damage, they would be under no obligation to pay for repairs.
All fuel companies mention that the calculations of your use are done in a software and include your previous use history and the average temperatures. It is possible that the new company botched the move of that historical data from the software the old one used and that's what messed up the delivery time.Â
Interestingly, I could never quite find a reference to run-outs due to the provider's miscalculation of your fuel use. They do usually claim no responsibility for run-outs due to your credit card being declined but your specific situation does not seem to be universally covered. This may be a little bit of a gray area that may give you some leg to stand on, and perhaps negotiate a price reduction for the delivered oil if you can get a proper person on the phone. I am certain the best you can hope for is a price reduction, not a completely free refill. Good luck!