I had worked for a company that changed its name. In my résumé and cover letter should I just use the new name or should I mention the old one too? This wasn't an acquisition or anything, simply a change in name.I guess one must periodically check the names of past employers to make sure of their name.
In a cover letter: "During my time at XYZ Corporation (formerly ABC, Inc)..."
In a resume: SuperAwesomeWorker, XYZ Corporation (formerly ABC, Inc), 2008 - 2010
I wouldn't worry about going too far down the list of entries on your resume to correct names of companies, unless it will cause confusion when people attempt to verify employment and find all the paperwork/emails/calls come from XYZ Corporation and you listed only the name ABC Inc.
In addition to jcmeloni's excellent points, I think it's worth mentioning that you should not worry about changes to the name of a company before or after you were working for them, just the name or names of the company while you were working for them.
There is absolutely nothing to be gained by updating your resume to say that you worked for XYZ Corp if it was only known as ABC, Inc while you were working there. In fact, it could be considered dishonest to do so. If you worked for a startup, which was acquired by (say) Google after you left, it would be a lie to say that you worked for Google.
It's actually a blank check to put any of the names, I advise you to put the most respected name, or the one helps you get the interview. And mention others in parenthesis after, or mention them in face-to-face interviews only.