560 (Q36, V31) - Should I retake the GMAT within 1 mounth?

It all comes down to whether you have sufficient time to study AND work on your app at the same time. I see no reason why you can't jump 40+ points in 30 days.

A few things to consider:
The GMAT is only "accurate" to within 30 points of your ability, meaning, if your ability is at 590, on any given test your score should range between 560 and 620. This is where some of the luck factor that people talk about comes in. Assuming your ability is 590 and you just had a bad day you could statistically have a 620 in you.

With regard to Knewton, I bashed them a little in my debrief, but in all honestly I think the course could be very valuable for someone aiming in the 600 to 650 range. Their teachers explain the basics very well, so that course alone might bring you up to your desired score. A word of caution: in order to qualify for the 50 point guarantee you have to view all their lessons, do all their homework and quizzes (which is a lot), and take all of their CATS (5 or 6 if memory serves). Given your timeframe and workload you might not get it all done in time to qualify for the guarantee.

I think a better bet for you is to pick up the 5 Manhattan GMAT quant guides, as well as their SC book. At a V31 you could see huge gains from the SC book alone. Their quant books are fantastic at teaching you all the fundamentals you need to score extremely well. I went from a Q25 to a Q47 just off the MGMAT quant guides (and 3 months of painstaking work). At the very least get the MGMAT Number Properties and Sentence Correction books overnighted to your door. The practice problems in the MGMAT quant books will direct you to the Official Guide 12th Ed., so you'll want to update your 10th Ed. to get the most out the the MGMAT books.

Best of luck!