I created a code tool that I can use through a VS Code extension to speed up how I work as a programmer. I have it in a private GitHub repo with an MIT license that I pushed up while I wasn't employed.
I'd like to use it at work, but I want to be able to take it with me no matter what job I'm at.
Would my current company have an IP claim on my project if I use it at work? It's weird because my code wouldn't make it into their codebase, it'd just be used while I code in their codebase
Not because of that, no. However -- depending on the terms of your employment -- your contract may have one of those nasty clauses that give them IP rights to anything you develop during your employment. If that's the case, they may have a claim based on that and that claim would apply whether or not the software was used at work. Those clauses are the first things I look for and negotiate away when getting hired.
My advice is to talk to an attorney about all this. A short consultation shouldn't cost a fortune.