What approach did you take to get your theme on the server? There are many potential approaches.
The recommended approach is to have your theme committed to a bedrock project. In this case, when deploy.yml clones the project files, it is cloning your bedrock with theme, from the repo you’ve specified.
As a different approach, perhaps your theme is separate from bedrock and available on your local machine. You could sync it to the remote similar to how the example project lists the theme dist files in project_local_files. In your case, you’d sync the entire theme directory, instead of just dist. If you take this approach, I’d recommend avoiding syncing large unneeded directories like node_modules. Create a file named .rsync-filter at the root of your theme, which lists which files to sync or not (see example).
I’m guessing you’re referring to updating your bedrock repo to the latest commit. With branch: master, your next run of deploy.yml should grab the latest commit from the remote repo (be sure you’ve pushed changes to that remote repo). If it is not pulling the latest, try temporarily setting the branch to the SHA1 of the latest commit, running deploy.yml, then setting it back to master.
If you’re referring to reprovisioning your server using trellis at its latest commit, I think you can just git clone/pull the latest trellis to your local machine, then run the server.yml playbook on the existing staging server. My impression is that recent changes to trellis are mostly just in how the playbook runs, not in the resulting server setup, so your resulting staging server probably wouldn’t change much.
If it fails or you run into trouble, you could copy out your db if you need it, then spin up a totally fresh staging server with the latest trellis, and load up your db.