Make sure to bring original expired GC and valid extension letter. Make copies of both in case if airline needs to keep a copy.
Print CBP carrier guide to have with you in case if you need to convince airline you should be allowed to board the plane:
https://www.cbp.gov/sites/default/files/assets/documents/2023-Nov/Carrier Information Guide ENGLISH.pdf
Show up to airport early. Speak politely but firmly to airline agent. Do not show you're uncertain or worried. Be relaxed and friendly or neutral. Be patient, do not interrupt them. If they don't know what you're talking about, show them the CBP guide. If they're not sure still, nicely ask for their manager / supervisor and explain why you should be boarded. Stay cool, polite, yet confident.
In case if you don't board the plane, look for a ticket with different airline. Consider flying to Mexico or Canada and crossing US border by land if you can visit any of these countries visa free or can get visa /eVisa / eTA easily.
Another option is applying for boarding foil in the US embassy inside the country you're stuck.
Most importantly, study rules of each country you're entering and transiting on your way from and to the US. Not all the countries will let you transit just because you have GC. A lot may depend on the passport of your country.
Good luck!