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Do not rule out being reunited with your passport so soon. Since you are going to fly to the USA anyways, if someone couriers the passport to you it just needs to get the courier's base right next to your starting airport. You will likely need a cab from the terminal to get to said base, in general there's no transit there. Of course, without knowing the exact airport we can't know for sure. In such circumstances 24 hours might be plenty, more often than not what takes time for the courier is to get the package from an airport to a residence. Do not forget to check the courier base opening times, often they are closed on weekends and such.