When I first came to the US ...
makes it clear that you are talking about something that is no longer the case and that something started the first time that you entered the US rather than on a subsequent entry. You are correct, however, that it leaves entirely unclear whether that something lasted for the first second, the first day, the first year, or the first decade. You can clarify that by explicitly mentioning the duration
In the first few months after I first came to the U.S. ...
tells us that the period is ended, that it lasted a few months, and that it started the first time you entered the US rather than on a subsequent entry.
There are lots of words in English. You are free to use all you need to make sure that you express your intent clearly. It may be good style to be succinct, but it is not good style to be succinct at the expense of expressing your thought fully and clearly.