I haven't seen them either and I don't really like the idea. However, I think your only bet would be to go to a Home Improvement Store's Plumbing Department/Aisle and pick out a few properly sized Plumbing Fittings with male threads to hopefully find one that fits your female hole (you know what I mean). Preferably, you want steel or brass, but PVC (thick white plastic) will also work, we both know you want something hard and stiff.
- You'd be trying to find differing spacing or tightness of threads. If or when you find a matching thread, then you'll need to slide the umbrella's shaft as deeply as possible into the also properly sized other end of the Plumbing Fitting. Then, it's time to drill baby drill. Drill through both the fitting and then through the umbrella shaft and out through the other side of the fitting for a well reamed hole ready for receiving a good screw.