I was in your situation once and everybody said the same thing to me: 'You really need to get some jet time'. It really annoyed/frustrated me at the time, but they were right.
If you want their advice (and mine), take a jet job in the RHS and have a bit of fun and then look for a LHS job/upgrade. That's exactly what I did and it was the best thing I ever did in my flying career.
A lot of people have tripped themselves up trying to take a short cut by going from LHS in a turboprop directly into the LHS of a jet. The vast majority of airlines of a reasonable repute will insist on jet hours before any move into the LHS.
Approach someone like Jet2 and say that you are more than happy to go back into the RHS and then after you have gained some jet experience, you would like to further your career and hope to be considered for a command. Most companies will react positively to that kind of approach. There will always be movement of crews in a company like Jet2 and therefore promotion prospects should be quite good.
I would suggest that they would be a good company to approach.
Good luck to you.
Last edited by 8eyes; 8th Aug 2013 at 08:15.