Easy to check headspace - you need a 30-06 "no-go" gauge. Any gunsmith worth calling a gunsmith should have one on hand and know how to use it (basically if the bolt will close with the no-go gauge in the gun it is not safe to shoot).

I would suspect headspace first. Other problems could be an oversize chamber (too big around), a LOT of crud or rust in the bore increassing pressure, etc.

If it is excessive headspace (probably is) it could have come about due to someone swapping the bolt out at some point or just due to wear. Also, although it has been a LONG time since I held a super sporter in my hands, I seem to recall that they locked up at the rear of the bolt - not really a good design for a cartridge as intense as the 30-06. Could be that the action has stretched over time.