If supplies are within a certain timeframe before expiration (give it 90 or 120 days, and there are countries setup to receive the vaccines and good to go. Ship it to them.
Start in our sphere (Mexico, Central America, South America and work outwards).
One could implement a law or rule that states if you go to the ER for non-COVID issues, if you aren't vaccinated you and the people dropping you off will be.
One thing though is J&J is basically is in this place given the manufacturing issues that they encountered. To me this is the best one to mass distribute given the easier storing of the shots.