Goldeswift20's journey

I currently work at Company A, where I'm sourced to Company B. My group lead at Company A is new and doesn't have much influence, so I'm considering asking my boss at Company B for a reference for a job opening there. It's common for people to transition from Company A to B, which contributes to high turnover at A. My boss at B has been instrumental in setting up a long-standing system that will soon be phased out, and I've been trained by one of the original employees for this task. There's a job opening at Company B that aligns with my skills and offers the chance to move to a different state, which I'm very interested in. I've already submitted my application, but I want to maximize my chances. I'm planning to research the organization to find out more about the position and show my interest. My concern is whether it would be appropriate to ask my boss at Company B for a recommendation, especially since she doesn't know I'm looking for other jobs. I worry it might come off as desperate or that it won't carry much weight in such a large company. I also want to avoid any awkwardness at work if I don’t get the position, as I know it could take a while to find a new job. I appreciate any advice on how to navigate this situation!