I have that type of corn too but treat it myself now and then with a medicated corn plaster. I have to be careful to cover the surrounding area with a plaster with a hole in it first, as the skin there is quite thin and the chemical in the plaster can burn it as well as the corn. It's rather awkward to do, but if I get the corn plaster on, and leave it for more than a week, including getting wet when I shower, the corn can be peeled off. Sometimes it needs a second treatment to get it all, but after that it will be gone for months - even up to a year.
I don't know why I get this kind of corn - I don't have any in the usual places on top of toes. I always wear comfortable shoes that don't rub, but this corn between toes is annoying.