That's a good question.
Regardless of the rules the chances boil to the following:
1) Do you have a contract with future rights or rights of first refusal attached?
2) Does the director like you? Does the director think you are the right person for the shepherd the production?
3) Does the producer like you? Do they the think you are the right person for the job?
4) Does the GM like you?
If the show is going to Off-Broadway (commercial or non-commercial run), there is no "experience" rule hinged on that.
If the show is going to Broadway, then there is the 67(J) Rule . . .
(J) In order to maintain the high level of professionalism in production as well as the necessary backstage safety and discipline for the efficient running of the production, the Producer agrees to hire as the Stage Manager only someone who has been employed previously for at least 20 weeks or in three separate productions as the Stage Manager, or 40 weeks or in six separate productions as an Assistant Stage Manager, or a combination of these (two weeks or two productions as an Assistant Stage Manager equals one week or one production as the Stage Manager) by an employer or employers maintaining professional standards at least equal to those maintained by theatrical employers who are members of The Broadway League, the League of Resident Theatres, the League of Off-Broadway Producers, The Council of Stock Theatres, the Council of Resident Stock Theatres, the Musical Theatre Association, Producers' League of Theatre for Young Audiences, the Producers’ Association of Chicago Area Theatres and URTA.
NOW . . . remember, if you are the SM, and the show moves up to Broadway, and you don't qualify, there is always the push for you to be the ASM.
But remember, if they don't like you there is ALWAYS some way to get you of the contract - buy you out, fire you before the transfer, etc, etc . . . there are lots of ways to work around the rules.
So, you are asking about the chances of you would get to the stay with the show, it all depends ........... NYC is filled with stories with SMs who have ridden their shows to the top, and also with SMs who got kicked out the show got "bigger". Interesting, there are also stories of SMs who rode their shows up . . . but soon realized (or it was realized) they couldn't handle the larger version of the production.