You did not "already pay tax". What you paid was only withholding, an estimate of the tax due. No piece of income is taxed individually. All your income, including enlistment bonuses, is added together at tax filing time. The tax is calculated on that total (after adjustments, deductions, exemptions and credits). Then total withholding is subtracted to determine if you owe more or get a refund.
By law, lump sum payments, like a bonus are withheld at a higher rate than ordinary wages. So, there is a good chance you'll get most (but not all) of that back.
