To the best of my knowledge (and being on H1B at one point of time), the H1B is tied to the company you work for because they mention it on the petition as well as the actual visa. So you need to be sponsored by the company you work for. The only time it is different is when you are sponsored by consulting firms who contract you out to other companies. For a full time job, you need to be sponsored by the actual company you work for.