What can I do to increase my chances of reclassifying to 31E?

You can't change the amount of class seats available when it's time to reenlist. All you can do is open yourself to other MOS's you're willing to accept. Application MOS's almost always have class seats and you don't have to wait to reenlist to reclass into them.

No matter what anyone tells you, there is no way to improve your chances to get the specific class seat you want. I've seen Soldiers call the school house, get recommendations from Colonels and Generals, have their leaders call the Commandants of the school, etc. Class seats are a finite resource and they are first come, first serve. The best thing you can do is check with your Career Counselor daily and immediately take the class seat when it appears. There are a thousand Career Counselors looking at the same class seat and they go to the first one who pushes the button. I've seen so many class seats and assignments go because a Soldier said they "would have to think about it" or they need to call someone. Minutes later that one class seat, or that single assignment to Japan, Guam, or Israel is gone.