You don't need the actual W-4. Just ask how many allowances you claimed. They should be able to tell you that over the phone without divulging any confidential information. It may also be on your actual pay stub, if you still have access to those. You can also find the same tables that they use in the IRS Web site and figure out how many exemptions result in the withholding you are expecting.
Note that the withholding depends on the amount of your paycheck and the frequency, so if you went from every two weeks to twice a month, you'd have a higher withholding. They also assume that you make that amount for the entire year, so even if you end up in the same bracket a higher paycheck would mean a larger withholding.
How do I figure out how this increase happened?
Download Publication 15 from the IRA web site. Use the filing status, pay frequency, taxable pay amount, and number of allowances, and look up the withholding in the tables at the end for each paycheck. Those tables should line up with the amount withheld each paycheck.