After I filed my tax return I got another 1099 in the mail for $5K. I already claimed this amount, but didn't enter it as a 1099. Do I still need to refile?

I assume it was a 1099NEC for your self employment?  No you don't need to enter the 1099 as long as you already entered the income.  Only the total goes to schedule C line 1.  Even if you get a 1099 you don't need to enter them.  You can just enter your total income as Cash or General.

No, you do not need to file an amended return. You said that you already claimed the income. As long as it is entered on the return you do not need to refile with an amended return. If you filed an amended return, not a thing would change in your return.