Is it possible to volunteer to deploy with a different unit than the one to which you're assigned, even if it is at another duty station?

No, if it were possible the majority of the army would all be deployed at one time.
The army deploys as a unit, and the unit trains together and prepares for the deployment as one organization. That process it takes several months to complete.
There are vacancies that are posted that you can volunteer to fill, but at your rank and MOS you won't qualify for any of those until you are a Sergeant First Class

The only time is when there is a shortage. This is really rare in the regular Army. They Army will send Soldiers to a deploying unit prior to their departure.

Sure, you can volunteer to do anything you want.
Buuutt...prepare yourself for a hard “no”. And then prepare yourself for some interactions like this;
PSG: ‘Private, raise your hand’
You: *slowly, nervously raises hand’
PSG: ‘Look at that! We got ourselves a volunteer! The sh*tters need cleaning, get going. Thanks, hero! That’s 6 days in a row!”
PSG:(mutters under his breath) “volunteer to leave the unit...good lord”

Seriously, what you’re asking really isn’t a thing at your rank. Best advice is to make your time in your unit the best it can be.

Best of luck.