I currently have the opportunity to spend a lot of time (meaning: not even a part-time job) thinking of a wealth of problems. This is great if I don't care about how I am going to handle my retirement (options: go homeless; die early; get adopted). I can fuss with details, spend time on the Internet researching or messing around, and still get enough sleep, time with family, etc.
You can lead your life as you choose. If you want to lead it best however, make sure that what you do is in line with your values. If you can get satisfaction from e.g. spending an hour a day thinking and writing math, schedule that in. If you absolutely want to integrate math into your current lifestyle, the major resource involved is time. Make sure that resource is spent wisely. (I've chosen to spend time away from math learning piano. We'll
see in a decade or so how good a choice that was for me.)
Under the assumption that you have approximately ten hours a week to devote to mathematics, I suggest building Internet and personal resources. Involving yourself on online fora such as math.stackexchange and MathOverflow may help, if you don't clash too much with the culture. Write up your observations in ways that might be understandable by others. Engage with other people, and establish relationships (face-to-face or long distance) with the understood purpose of mutual mathematics enhancement. If you can get a mentor, even better. Find out if you like to solve problems ("repair work"), build bodies of knowledge ("catherdral or bazaar builder"), or note interesting connections (synthesis). If you can leverage your existing skills to the process, so much the better. Get to know your research library. If potential mentors have offered up their email, use it; don't be afraid to ask, and always respect the others' time.
Above all, manage your resources, primarily time, energy, and satisfaction level. Hopefully nature will tell you which direction to follow, and a support group will tell you (what not to do) to get there.