If this is a modern world, how about a little almost-technomancy? I assume if there’s a mad scientist capable of sewing two heads onto a single body, there could be a second skilled mad scientist/tech whiz who would be up for the fun of hiding two heads in plain sight. There’s two science-related solutions I can think of off the top of my head.
1) The second head can be cloaked by a cloaking device. This one would be very hard to achieve with modern-day technologies, but perhaps in the future it wouldn’t be nearly as difficult. Even in the modern day we have some more limited prototypes capable of certain ‘stealth’ effects (see here). Whatever cloaking device you use, you could hang a lampshade (not literal) on it by explaining any technical glitches that happen with it on makeup, special effects, or covering it with clothing. The device would probably work very well in dark places, like clubs or some bars. It would also make the story more entertaining because the flaws of the device itself affects how your character interacts with her world, eg. hiding in shadows, an occasional baddie seeing through the disguise and wiping his eyes in confusion after seeing it back to normal, etc.
2) We make the second head somebody else’s problem. This refers to a device from the book series Hitchhiker’s Guide to the Galaxy, where a Somebody Else’s Problem field is generated to make people ignore whatever is under it, as if it’s ‘somebody else’s problem’. This one is more of a hard sell; essentially there would be some kind of hypnotic device that induces others not to pay attention to the second head. You could again use makeup, clothes, etc to make the other head even more attention-worthy, especially if it were a head-turner. This one could also make for fun in the story, as glitches can make things more adventurous, although if the device were glitch free enough it could also allow the character to be more carefree in their interactions. It might even be good enough that people touching the second head would have no idea. (It’s also, in my opinion, the coolest device in the Hitchhiker series).
Edit: 2.a) This one is also out there, but on the subject of clothing, perhaps as your character is out in a crowd and needs to stay out of attention, a piece of clothing with some optical illusion patterns would be nice. It could let your characters head seem smaller, or, even better, give a severe headache to anyone who focused their gaze on it, enough to let them overlook the ‘slightly larger’ head of your character.