Half my FO's lately have been turboprop CA's from other regionals. I'd say you need glass/jet time and some 121 would help too. I'm guessing that 121 jet SIC would do the trick since you can already check the TPIC box.
I suspect 121 time would be better than 135 if the goal is the majors. You'll meet more people who are major-airline bound, and get to talk to pilots from various airlines. Also good chance that majors like previous 121 experience since it's similar to the schedules they fly.