I used to know a guy with a lamp on the top of his helmet on a horizontal swivel. It rotated freely so the light would stay level no matter how he tilted his head. But it didn't rotate left or right, so it always pointed in the same direction as the helmet on that axis. It was a rear-facing red light - I don't remember if it also had a front-facing light. I had a conversation with him while he was wearing that helmet, and I couldn't take a single thing he said seriously with that light swiveling constantly.

Recently I met someone with turn signal lights on their helmet, controlled by a wireless transmitter mounted on a handlebar. It's not exactly what you're asking about since I don't think it had an always-on light for visibility. But I mention it because I thought it was pretty cool.

I use a high quality camping headlamp that isn't mounted. As well as a headlight and a blinking light.

Headlamp to see when I move my head.

edit: linked to the headlamp I use.

I have an older light that came with an elastic webbing arrangement for putting it on a helmet. I have never used it, so I can't say how well it works.

You can probably just use any handlebar mount that uses a flexible band instead of a rigid fixture. I've personally used this, but anything similar would do.

I use a headlamp with straps for biking. The kind that have the top strap for like hardhats or whatever are good for bike helmets too. That way I can see where I'm looking and let the bike headlight shine at cars. I buy the cheap ones from harbor freight.

Mine is built into the helmet.

Depending on your helmet, you might be able to find a 3d printed adapter that fits in to a hole on your helmet. Something like this: https://www.thingiverse.com/thing:426012

Helmet lamps are the perfect way to constantly blind yourself in foggy conditions. Or rain, or snow.

These Bontrager light clips just tie through a vent hole.

https://encrypted-tbn0.gstatic.com/images?q=tbn%3AANd9GcTSMr5xX7Ku5zx3jQL0HQ2L60SpTV0oza7IKl_4DwU-uw&s=10

It allows me to run flashing DRLs during the day and evening hours for cars, or bright lights at night for the darker multi use paths. Pretty good. Also, pedestrians will constantly ask you what camera you have on, and you'll constantly have to say "actually, it's just a light" hahah.

Onto the back of my helmet because I don't want Ebeneezer in his 3tkg grand wagoneer to convert me into paste.

Just screw it into your helmet with a self-tapping wood screw /s

Urgh, I hate people with helmet lights or any blinding lights in general. Today it's not only huge cars blinding everyone in their way but also bikes with way too bright LED headlights that are badly adjusted. And to be clear I almost exclusively travel by bike. I find it way more important to watch and be aware of my sourroundings.