L1 to H1B : Change employer with H1B approved

I am currently on L1b visa, with h1b approved for consular processing. I am working with current employer for change of status from L1b to h1b. Should I wait for H1b COS to be complete, before working for new employer? What is the best option here - Wait for H1b COS to be complete with current employer or let the new employer take care of H1b COS.

Either is fine. It is your choice. One would be filed as COS, another would be just Transfer. So, it does not matter…If you have an employer willing to file to COS and Transfer, you can ask them to file it in premium processing and start to work for them after the approval comes in. Applying two COS at the same time by both employers can be tricky with the timings, so take one path and stick to it.

Thanks for the response. Which is a less risky option ?
Also, I have never applied to any other employer in US so far. How common is employers filing COS vs H1b transfers ?

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Getting COS approved with current employer and then applying for Transfer.
It is less common…but, happens though…