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Is there a more tactful route I can take that doesn't involve (as
much) lying?

It is none of their business, do not lie about it.

Lying is never a good idea, and almost always comes back to bite you. I would urge you to simply say "I was not feeling well, but I feel fine today." and then change the conversation with another line like "Did I miss anything yesterday?"

There are plenty of 24 hour illnesses that can be used to fill up the rumor mill, but I would not worry about this. Stick to the line above and do not elaborate.

You do not have to disclose anything. But, if you just don't tell anything, everybody will suspect their own thing and it can start rumors. So this is one of the edge cases where I would suggest a harmless lie, as it is the best for all involved:

One day => terrible headache!

There is nothing more to tell about that, there are no after-effects like with a cold etc. and you can even prime the next event by telling you sometimes get those when really stressed.

I am in a similar position where I've had to take several days off to attend counselling sessions and to see my GP. The only person at my company who knows - as far as I know - is the HR department. The people I work with don't know.

I have had time off, and I've just said I've gone to the hospital for check ups. One person asked more questions about it, and I simply said that I appreciate the interest but it's not something I want to discuss as it's private.

"I had a medical condition flare up yesterday. Usually it's fine, and it seems fine again now. Yesterday was just an odd day, rather unusually bad. It's good to be back."

This actually gives a lot of details about your personal experience, without a trace of the "mental" aspect of the situation. Since you are regularly taking medicine, that doesn't sound dishonest at all.

If they press:

"Truthfully I prefer not to get into those details with co-workers. I'm fine now, though, and expect to be good for some time."

Here's a way of telling a coworker that you needed a time off for mental health-related reasons, without revealing the "mental" aspect of it.

It's a strategy I used myself, and it worked out great.

I have a neurological condition that is usually not a problem, but
at times it may become prominent and requires my attention. But it's
nothing too concerning. I'm alright. Thank you for asking!

Perhaps replace "neurological" by "health", at your discretion. The key here is not mentioning "mental".

Presumably, your condition is a neuropsychiatric one, so you are not lying.

If your coworker still asks about what it is, exactly - which is a bit unlikely, since you implicitly avoiding being too specific should be taken as a hint - you may just say something among the lines of:

Ah, I don't feel like talking about it right now, I'm sure you
understand. But no need to worry about it, really.

In fact, when I said this after returning to work at my previous workplace, one of my most intimate colleagues asked me, in private, if I'd mind telling him specifically what problem it was, and I saw no reason not to. In my case, it was major depressive disorder, which he happened to know about, more than I anticipated.

It should be enough to just say "I was feeling unwell, but I'm doing better today!"

If a coworker presses the question and you don't mind sharing a little, responding with a partial truth like "I didn't sleep at all, but I managed to catch up somewhat" might be enough to assuage your coworkers' curiosity. (But I'd avoid saying anything false!)

As a general strategy, if you want personal details to stay personal, try to avoid sharing personal details in other situations. If one of your coworker normally sends detailed emails like

I'm down with TB, expect me back in 30 days o.O

or

Both kids are sick with stomach flu. Better up than down at least! Back tomorrow!

then sending a mail like

Sick. Not sure if I'll be better tomorrow..."

will probably invite curiosity. (This strategy applies to other personal leaves -- if don't want to reveal that you're taking a personal day to marathon both LOTR trilogies, don't broadcast details for other vacations you're taking.)

I'd suggest spinning it as "I had a severe headache" (or migraine). It fits right along the lines of something in your brain/head area that was affecting you, rather than revealing the true reason. You're also less likely to show 'tell signs' that you're lying, as opposed to claiming something completely different like "I had a house fire".