New passport with name correction

[Chamiya]

Hi,
My passport, visa (H1) and I-94 has only First name on it.

Now that my old passport exipired and got a new passport with First name and Last name added to it.

I am going on a month vacation to India on April. Will there be any issue while coming back with my old visa, old I-94 on old passport ?

If so what do I need to do ? Please advise.

(I asked few of my friends and they say as long as I carry my old passport and new passport together, that would be just fine)

Thanks,