How to move via layer 2?

Say I have an ERC-20 token I want to move to an exchange to trade but don't want to pay Eth's massive gas fees. The token I have didn't seem to be on any CEX that support any layer 2 protocols and most DEX don't support the token. The ones that do seemingly just swap between tokens instead of have an actual exchange for trading.

So how do I go about moving this token to an exchange via layer 2 if both of the above are true? Do I just eat the loss and pay the Eth gas fees until DEX get better or CEX support layer 2?

Token in question is JASMY.