On the tax form there is only a space for a middle initial, not the full middle name. And it doesn't matter whether you include the middle initial or not. The IRS doesn't pay any attention to it.
You do not have to tell your employer to add your middle name. Even if you do ask them to add it, they certainly should not issue a corrected W-2 to add your middle name.
You do not have to coordinate changing your Social Security card with filing your tax return, as long as your first and last names remain the same.
The automated matching done by the IRS computers uses your Social Security number and the first four letters of your last name.
