My personal experience suggests that reaching reasonable breadth of mathematical scope is achieved through three different mechanisms :
Attending talks (seminars, colloquia, workshops) in subjects where you are not an expert. Not only you will eventually grasp some useful bits and pieces of how to think about various mathematical phenomena, you might even find some things useful years later when you encounter something similar in your own research.
Trying to browse arXiv regularly. It is getting increasingly difficult, since the amount of daily postings is much higher than 15+ years ago when I was a PhD student. But still, scrolling through titles of papers and occasionally reading the abstract and/or the introduction when something catches your eye can be very useful. A related advice: if you read something useful and/or exciting in a paper, check on MathSciNet what papers cite it; this might bring you to some other interesting things.
Try to find a way to interact with your supervisor to broaden your scope. Good supervisors would only limit themselves to a narrow subject of your intended area if you insist on that. Having a mathematical mentor who can share with you his or her broader vision of the field, of interesting things to learn, of interesting events to attend, is perhaps the greatest thing that PhD studies offer. Use it wisely.