A valid Mexican passport "should" be fine for the OPs question.
Originally Posted by
reclusive46
TSA don't routinely do any kind if immigration checks either.
Until they do; overzealous agents have been known to bring up immigration issues to LEOs- it is rare, but not unheard of.
Also, while not normally at the airports in the OP's question, CBP often stand right at the TSA doc check station and do immigration checks at border airports like ELP, BRO, BUF, etc. I would not suggest one travel from those airports without proper immigration papers; CBP can and does detain people at the TSA line.