I am sitting here in a big jet avionics shop talking with my co workers talking about your problem. Assuming your aircraft has a DG for the DI (HSI), the most likely culprit would be the DG. Possibly vibration induced or recessing (only a coincidence), once the DG gets up to speed to supply a heading platform the forces it is exerting may be actually stabilizing it. One Idea was a combination of the DG spinning up combined with voltage increase as the engine was brought up to speed.
Just keep that GPS backup with fresh battery's
More help if we had details on your avionics package and the age of the aircraft.