As Lori mentioned, it would be a good idea to check with various operators in your area and see what they are looking for not only in terms of flight time, but in terms of background, character, etc.

With 350 TT, you're not going to be very helpful in the cockpit. Sure, you may do even better in training than someone with 2000 hours but you'll still be quite a bit behind when you're actually on an approach in the ice and blowing winds encountered in many places in the U.S. during winter. All that said, you may luck out. You wouldn't be the first one. More than likely however, it'll probably take 1000-2000 hours or even more before you'll sit up front in a turbojet. But as I said, I've met multiple pilots who were flying jets with 300-700 hours.

The CFI is a great rating and has brought me many wonderful experiences. You also get to meet many people you might otherwise never have a connection with. I'd highly recommend working on that rating regardless of whether you pick something up at 350 hours.

Mr. I.