Mexico only requires a valid US passport at time of entry. Most other countries do require the 6 month requirement and the airline may hassle you but once they check Timatic it would be resolved.
Using an expediter service or the same day urgent service (for additional fee) if you live near a passport center may work for you. For that you have to prove you have international travel already planned and ticketed. Be prepared to spend several hours in line for that.
Letting your passport get less than 6 months is a risk. You should institute a renewal as it generally takes several weeks unless you're willing to pay for expedited, expediter or urgent travel services.