i guess it all depends on what program you are looking at. if you think you underperfomed than do it again. i was looking at it from the perspective that "he took it twice, got basically the same score twice so he is scoring at his level". Your time on earth is finite, and the obsession with the GMAT has to end at some point and people have to move on.
I would say most adcoms would have a similar thought process on first glance as well. unless you want to use your optional application essay to tell them that you took the GMAT the first time totally unprepared (as you can imagine, this won't impress B-Schools) and then tell them the second time you did not do well because of poor time management (again, you aren't going to win anyone over here either)
most programs i've looked at have an avrage around 700. i've seen some good programs that want you to have over 650 (McCombs??) and even with averages of like 680 (Maryland?) Of course i've seen lots of averages with a 720 etc (NYU CORNELL.) if you think that the higher score is what you need to differentiate yourself go for it. Personally, I just don't think a 20 or 30 point - hell even 40 point - improvement on your *third* try is going to wow anyone on a PhD committee.
I don't think GMAT score is the important deciding factor you think it is for PhD programs. Average GMAT scores are so high because really talented people go for their PhDs... yes of course... so you need to score high and should aim to break 700. but remember, PhD programs aren't subject to ridiculous rankings by magazines like BW. (yeah I know FT does a ranking, but no one pays attention to it) PhD programs do not make money for Universities like MBA programs do. PhD programs are about researching and producing knowledge.
If applicant X has a 680, but worked on a series of published papers as an RA he will beat the egghead with a 750 GMAT and no experience any day. If thats not the case, then its not a program that has their priorities straight, and they aren't interested in producing high quality PhDs who actually MAKE IT THROUGH the program (PhD is worthless if you don't complete it)
You have a professor on your side here, do as much work as you can for him. If I was in your shoes, not a precious extra minute would be spent on the GMAT - I would be groveling at this Professors feet for any bit of work I could get my hands on.
That's just my opinion. Don't take offense to anything I said here... everyone has the path that is right for them...
P.S. what Top 20 MBA program does not have GPA? That is sounds cool/interesting