A few months ago I started a thread
talking about the issue of job applications that ask for "previous names" or the like, and the legalities of asking the question (turns out it depends on when in the hiring process it's asked and how it's worded), but didn't discuss how to handle it in practice if you encounter it.
First of all the fundamental reason for asking it is so they can properly check relevant records. The recommended way to ask it (speaking from the company's perspective) is thus "Do you have any ([insert here] work, school, criminal, etc.) records under another name?" That enables them to seek what information they need - and no more. Unfortunately you can and probably will run onto the question asked in a more gray-area form, like about asking for any other names you're known by. The good news is depending on your situation you may be able to get around outing yourself without lying by answering in a roundabout way (by the way I do not advocate lying, but rather using alternate ways of answering the question and/or determining what they need to know). These "alternate methods of responding" are based on advice I've seen regarding other touchy and/or illegal inquiry subjects (e.g. asking about your family, whether or not you're a citizen, etc.) and responding in a way that answers what they need to know yet doesn't lie (mentioning you can meet the requisite work schedule, that you're authorized to work in the country without restriction, etc. respectively).
If you don't have ANY records they'd want to check (either because you transitioned young before you ever worked, had any credit, etc.) or you've updated ALL your educational, employment, and credit records, and don't have any criminal history under your old name (if you do this tactic is basically gone out the window forever), you have an option I recently thought of: When asked, simply state something like "none that any relevant records would be under". (If you can't honestly give that statement then this strategy won't work.) If they ask what the deal was, and you've changed the name (not necessarily your gender if you haven't done or can't do it) on your birth certificate, you can mention that there was a change made to your birth certificate. Chances are they'll assume you were adopted or had some other childhood name change and will say they don't need to know said name, and thus you will have avoided outing yourself (to be honest I think that most non-TGs in the aforementioned categories don't even bother mentioning that or their birth name and they don't usually end up getting accused of "lying", but in a TG's case with employers finding ways to cover their discrimination it's best to be sure first). Obviously this idea probably won't realistically work for those who transitioned much past the start of adulthood as it will likely be too difficult or impossible to ensure everything (and I mean everything) that the employer may want to see is under the new name.
So what can the other TGs do to avoid outing oneself to an employer? If some records (whether it be work, school, credit, criminal, or other) that they want to check is under your old name, you probably don't have a choice but to out yourself (provided that the check they're doing is legal). What you might try doing if you can't honestly use the strategy in the last paragraph, but what they want to check under your old name is something that chances are they won't go after (like your high school records once you've graduated college or work history from decades ago), you might try putting something down like "will mention/discuss if needed"; unlike the tactic mentioned above this one may raise more questions though.
One caveat: What I say applies to private-sector jobs in the U.S. If you're applying for a government (especially Federal as they're exempt from the normal suite of "illegal questions") job, or a military, police, etc. job, you probably will have to mention your former name regardless of whether or not it's relevant (and especially so when applying for a security clearance).
Another side note: Although admittedly TGs are some of the most severely affected by these kinds of questions, being asked for former/other names you're known by without the question being tailored to the specific purpose like I mentioned, could raise legal issues with regards to other groups too if a pushy employer considers not mentioning said names to be lying (except in these cases where an employer is trying to find some excuse to fire or not hire someone I think most of the time companies see the old names question as more of an "informational" one like asking for your phone number or e-mail address rather than an "under the oath" one). Examples of other affected groups are (like I've mentioned before) immigrants who change their names in order to assimilate, and beyond legal name changes those who use pseudonyms for lawful purposes like authors, those who blog under another name, or those who have a name used only in religious circles; it would especially be legally an issue if the subject of the book/blog/etc. is something about religion, one's national origin, etc. Then again, if those people know about it ahead of time, they can employ the same tactic I mentioned above of saying that no relevant records are under the other name and escape unscathed without lying.