Do you have an IRA? If so (and you haven’t hit your contribution limit), put the extra money there. If not, open one (you can easily do so online) and tuck the extra cash in there. I suggest avoiding banks for this one; their investment options are usually terrible.
The nice thing about IRAs is that they follow you your entire life; you never have to wait to have your new job set one up, and if you find yourself unemployed or in a job with no 401(k) plan or you max out your 401(k) contributions, you can still contribute to your IRA.