Not lots of good choices that are easy.
The .22-250 bolt face in your 788 is the same as the .30-06, but in a short action. Technically, you could do 25-06, .270, 280, .30-06, 338-06 (my personal favorite) or .35 Whelen. There are others, but those would be the mainline calibers at the .30-06 length. To make them work in the shorter action, you would have to relieve a slot in the front of the ejection port and re-work the magazine, but it can be done.
I assume you are not enamored of the parent to the .22-250, the 250-3000? You could go for a .25 Improved...
Did I say I think you should go with the .338-06 as a lightweight hunter?