Hey, Tuck.
If you don't want to take the GRE again (I don't blame you! No, really, I don't), then you'll really have to shine in other parts of your application. One professor I spoke to from Northwestern said that I needed to take my GRE again while others said my GRE scores were fine (I scored above 80th %). In one particularly frustrating example, a Vanderbilt department recommended me for acceptance to the dean and the dean turned me down because of my GRE score. So, there's even debate about what is acceptable within school administrations. It's just so variable. Higher GRE scores certainly don't hurt your chances. You may want to think about taking it again.
I do find that fit is the most important aspect of any competitive application. Of all the criteria, only fit will make you or break you (unless your GPA is 1.5 or your GRE is in the 140s). I have no idea what your stats are but I am willing to bet that they are competitive. So, I would focus more on really honing in your SOP for each school you apply to. I spoke to several chairs of SW PhD programs, and each and every one of them said the same thing: wholistic review of application materials but you have to show that you have really put thought into why a research degree in SW is rooted in both your personal history and career ambitions. Did your SOP describe exactly what you wanted to research, how that research could be supported by the faculty in the program based on their expertise, and how your research/career ambitions are deeply rooted in your SW worldview?
Beyond the SOP, I don't see why some stats/research methods classes would hurt, especially if your GRE quant score is below the 60th-70th percentile. Do you have any publications? What was your writing sample like? The programs you applied to (If I remember correctly) get the most applicants. Applicants who have publications, above 80th percentile GRE scores, outstanding LORs, graduate school GPAs above 3.8, research experience in applied labs, and are racially/ethnically diverse.
[Side note, both Upenn and Berekely professors asked me if I am a white male over the phone. When my answer was yes, they both followed up with, "well, what makes you diverse?". I explained that I am gay and they both (different phone calls) let out a collective sigh. This surely points to the pressure that these programs put on themselves to keep their student population "diverse", however they define diverse. It also points to the nature and status of the SW field. By the way, the professors were both white males. Oh, the irony.]
This is to say that there are somethings in your "application" that you cannot change. Applying again also means the applicant pool will be different. Perhaps measured up against different applicants will help you, aside from the fact that you will be showing grit/determination/loyalty by applying again. But again, I really think it's about fit. I've received interviews from Columbia, BC, UW-Mad and I have to say that the overarching theme is that they would be a great fit for me out of the 11 programs I applied to.
I don't know. Just my 2 cents.