You can't get "a refund for the tax I paid on my unemployment compensation" because, by omitting the 1099-G from your tax return, you didn't pay any tax on your unemployment. (If tax was withheld from your unemployment, it was applied to the tax on your other income.)
You have to file an amended return to report the unemployment benefits. The first $10,200 of your unemployment will be tax-free, provided that your AGI, not including unemployment, is less than $150,000.
