This thread is in the wrong place, needs to be moved to the IMG forum.
I assume you're completing a PGY-2 prelim position now, and you didn't match this last march to anything. Clearly your best option is to try to find a surgical PGY-2 or PGY-3 position outside the match. Alternatively, you could look for a PGY-1/2 position in other fields (PGY-1 IM, PGY-2 Rads, etc). All depends on your interests, etc.
If you don't have any clinical position at the end of June, then you'll need a new visa. Which visa you can get completely depends upon the position you get. You might be able to shift into a research J visa. You mention a "fellowship", but here in the US you usually have to finish your core residency before you qualify for that (although there are some non ACGME approved fellowships that may be more lax about it, but they usually do not qualify for J visas).
If you're asking "can my J be extended while I look for a new job", the answer is definitely no. The J visa is tied to your training position. Once your position ends, you have 30 days to leave the US (so there is a short grace period). If you want to stay in the US, you'll need to find a new position, and then get a new visa (unless you find a new clinical training spot, and then you'd just need a revision of your current J visa). Since it's June 1 today, the timeline for finding a new position (and possibly a new visa) is very short. Being "out of status" (i.e. in the US without a valid visa) can be a big problem given the current administration and it can make it more difficult for you to get a visa in the future.