I'm installing the last 100 sq ft. of about 2000 ft of solid 2 1/4" natural hickory.

Most of it has been in about 18 months, but got I got sidetracked before installing the rest.... In any case, the stuff that is already installed is stapled to joists every 24 inches, with staples in between at the recommended schedule from the manufacturer. The subfloor is 3/4" OSB. The portion that's installed is holding up nicely, no noticeable looseness or squeaks.

Now, the problem. The final room has joists running parallel to the direction the flooring needs to go. I'm not sure why the builder chose to change directions on this small section, but he definitely did; as the basement is unfinished and I can see the joists from below! This room and another are separated by 5' wide doorway, with no doors, and it will look funny to change the direction of the flooring at that point. However, by installing parallel to the joists, I won't have nearly as much strength, since OSB isn't especially strong.

My question is, since the bottom is exposed, is there any advantage (or harm) to clinching the staples (bending them out to the side) after installation?

Another alternative I thought about is putting in a few cross-pieces between the joists, but since the joists are engineered, they're not especially easy to fasten to.

Any advice would be appreciated.

Cross bracing would be best, one every 16 inches will do the trick. You can use joist hanger brackets to make it a bit easier.

as far as bending the staples, in my experience only about 1/16 of a staple will actually go through the subfloor. Hardly enough to bend

I've got about 3/8 to 1/2 on most staples, but cross braces seem sturdier. I hadn't thought of joist hangers, that might make it easier indeed.

Thanks!