H1B visa transfer and travel

I was working in USA for 3 years and due to covid issues in my family, I travelled to India during March 2021. I had my passport stamped in November 2021. Since the Visa stamping process took so long, my client and employer moved me to offshore payroll. Now to return back, I would like to know Is it fine for me to travel with my current valid visa even though my employer is not moving me to onsite payroll and move to a different employer after I enter US. Will there be a problem during immigration if I do so? Or should I file a petition with another employer now itself(in India) and I can travel ? Please clarify

You should have a valid job offer from new employer in the US on the day of entering the US.

Your payroll should start immediately on day 1 in the US

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So in that case, with my current Visa I can travel without any issues in immigration? If I find a new employer being in India , do I have to notify in the immigration that I have a petition in processing? Please clarify.