Working for the old company for a week after joining the new company

[ashishmgupta]

I have H1-B visa with company A (full time job). I have resigned from company A and my last date has been finalized. The H1-b with company A is valid. Also, my H1-b has been transferred to company B (full time job) and I have got the approval notice from USCIS. I have promised to join them on a given Monday and my last date with A is previous Friday. However, for the experience letter, company A is asking me to work for 40 hrs in the week I am joining company B. I am okay to work after hours and weekend to put in 40 hrs only for that week without pay to get the experience letter.

Questions :-
1) Would that be legally OK to work for the 2nd company and also working unofficially for company A (after hours and weekends)?
2) If the company A includes the week i am working for them in the experience letter, would that be a problem in the future (for the green card etc.) because officially I am already with company B?
3) Should I ask company A not to include week I am working for them in the experience letter?

[immihelp]

It is up to you whether you want to work for free for one week for them. However, from their side, they are obligated to pay you for every hour you work.

If you are not getting paid, it is best if they don't mention that you worked for them during that time.