if your immigration status is OK

then the only thing left is if any of these codmpanies have policy that preventing work for other companies in the same industry. If they are in the same field, you better be carful.

It is not a good idea to talk to HR because no company will encourage employee to work for more than one full time job, not mention two.

One of my friend did that though for a short period of time, (6 months). not same industry.