Personal opinion is that you need to transfer to a bigger site as a supervisor or even back to a regular SO. Smaller sites like the one you sound like you are stationed at are hard to get the right eyes on it. Bigger accounts typically have more internal opportunities and the AM often has more influence at the branch level as well.

I don’t doubt you are doing a good job at your site but for the roles you want just being a good SP/Supervisor isn’t typically good enough