Ways to spend gap year/semester before applying to mathematics PhD
Options I can think of (I bet there are even more than these):


Take some upper level undergrad, or first-year grad, math classes as an unmatriculated student
Audit some classes (this saves you some money -- you don't get credit, but you learn the material)
Apply for a master's, in a program that doesn't require a math GRE (for example, with some quick googling, I found a good program in statistics that doesn't require the math GRE)
Take or audit some non-math classes that you find interesting
Become a K-12 substitute teacher (this only requires a Bachelor's degree)
Be a Vista volunteer
Get a job and build up a bit of savings, to make it easier to get through grad school on a TA stipend
Study abroad for a year but not as part of a PhD program for the reasons you mentioned


Note, if you are working full time, I find that one challenging class at a time is enough.  If you are working 20 hours a week, depending on the type of job, maybe two challenging classes would work comfortably.  I remember taking data structures and linear algebra at the same time while I was working a 40-hour week, and I went a little nuts.  In retrospect, it would have been better to negotiate a reduced number of hours at work.