You need to have a part-time H1B with your advisor.
If your H1b with your advisor still have more time, have a lawyer file a amendment to change it to part-time.
If your H1B with your advisor is going to expire soon, have a laywer file a new part-time H1b for you.
Either case, keep a fulltime H1B with youer new employer and part-time H1B with your advisor.