In an arrangement with the US Government it could be put into the ground, elsewhere. Doing it without state permission would be an entirely different matter however. I'm not aware of any very large, privately owned underground storage caverns (such that they could make a dent in absorbing the over-supply).

This is being discussed:

"The U.S. Energy Department is negotiating with nine companies to rent about 23 million barrels of oil storage capacity in its Strategic Petroleum Reserve as part of a Trump administration bid to help drain the country’s growing glut of crude."

https://www.worldoil.com/news/2020/4/14/doe-working-to-lease...

Here is what the US Strategic Petroleum Reserve looks like (it can absorb some of our over-supply briefly):

> Four underground salt caverns on the coast of the Gulf of Mexico store the oil. That's a central location. The oil can be distributed to nearly half of the U.S. oil refineries either through interstate pipelines or via barges. It only takes 13 days for the oil to enter the U.S. market from the time the president first gives the order.

> There are two cavern locations in Texas. As of September 30, 2018, Bryan Mound held 235.3 million barrels in 20 caverns. Big Hill held 153.4 million barrels in 14 caverns. The other two are in Louisiana. West Hackberry held 199.5 million barrels in 22 caverns. Bayou Choctaw held 71.88 million barrels in 6 caverns.

> The maximum capacity of all four caverns is 727 million barrels.

There is also a good FAQ on it:

https://www.energy.gov/fe/services/petroleum-reserves/strate...