You will get some of that back.
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,enlistment bonus, is added together at tax
filing time. The tax is calculated on that total (after adjustments,
deductions, exemptions and credits). Then withholding is subtracted to
determine if you owe more or get a refund. The enlistment bonus is already included in box