If it is only one month of overlap, there won't be any significant implications that it would be worth worrying about for such a short time.
Generally, if you were to be working two or more jobs at once, you would want to make sure your payroll is withholding a bit more taxes than they normally might withhold. This is because a lot of payroll is built around assuming that an individual will have the standard deduction and personal exemption on the first $10,350 of income. When you have two or more jobs, each job will be assuming the same thing, but, in reality, you have more income. But, for one month, it is not worth worrying about or trying to change your W4 to withhold more taxes.
Procedural wise - your tax return will be filed out the same way, he will just have two W2s. So, make sure he remembers this next year, as both companies will send him W2s. Its very important you wait until you received all your forms before you file.
