Look at my resume, and you will see your story is pretty much my story - after six years of classical theater, and I am trying to shake it up a bit - for a wide variety of reasons.
It's important to sit down with a director, and sell yourself. First know, that it is often easier for a straight play stage manager to transfer into musicals then the other way . . . It's odd, but it seems to be true.
Sit down with a director, general manager, producer, and point out the selling point of Shakespeare . . .
1) Larger cast
2) Ensemble casts / Chorus style tracking
3) Fight /Dance / Music in Shakespeare
4) Multiple Scene Changes and locations
5) Large Artistic Teams (Director, Fight Director, Vocal Coach, Musical Director, Choreographer, Etc, Etc)
AND . . . don't forget, play to strengths, there are people out there looking for good Shakespeare Stage Managers, so maybe opening up where you work. There a lot of people out there would love the opportunity to work Shakespeare.
I have been using connections with directors and theaters to get into new work. (For example, I got a job a new theatre by doing Shakespeare which lead to my impressing them, which lead to working on a new musical . . . )
Good luck.