Congratulations on your new score! A 700 will vastly improve your chances on admissions. I was going to respond on your 610, since a lot of other posters seem to have a false sense that a low 600 score will make you competitive for a top 50 program (it most likely won't).. These posters are either thinking you're applying to an MBA, or they are not aware that admission into a PhD program is far more competitive than admission into an MBA program. With a 700, the top 50 (for the top 20, generally the scores need to be even higher, perhaps 730-750 range) should be more readily open to you, as long as the rest of your application is good. I would say that a 700 is probably generally considered the bare minimum in making yourself fairly competitive for the top 50, although school websites may list lower figure.
I would hesitate to mention teaching experience as good work experience, especially for applying to the top schools, as I think that generally teaching work experience is not very valued. In fact, be very careful about emphasizing your teaching in general because, believe it or not, that can get you automatically removed from consideration at good PhD programs. Work experience in general is heavily discounted in the PhD application process (although great industry experience can work in your favor).
There are other factors in getting accepted, such as your chosen discipline. If you were interested in a PhD in finance for example, even with a 700, I would question whether your application might be competitive for the top 50. If you'd like us to provide a more complete evaluation, you could give us some more information, such as your GPA, what did you get your BS/MS in, what discipline you are interested in, why you want to pursue a PhD, whether you will be able to get strong letters of recommendation from research active faculty, your age, what math courses you have completed, what your verbal/quant breakout is, etc..