5/16 is certainly a big enough diameter. If the holes are 3/8 then simply use washers. And you SHOULD be using washers in any event.
If you want to get serious about centering the 5/16 bolts in the holes put a few turns of electrical or masking tape around the bolts just under the washers to center them.
Or simply sigh and leave it until tonight when you can check the press?