Do you have any advice for an E6 moving towards E7 (e.g. ASI, PME, College, etc.)?

Back when I made E7, I came out number 1 on the list for my MOS, and from the secondary zone. Everyone in my MOS looked similar in regards to ASIs, PME, secondary MOS, etc. I had a college degree complete (prior to enlisting) and most on the list did not. The same was true for the E8 and E9 lists.
Finish your college degree.

Seek broadening assignments such as staff, special duty (ie recruiter, DI, IG), and institutional training. They will serve you well in practical application as well as give you a leg up when competitive promotion boards come around.
If you haven't done so yet, make sure you get some troop time as a squad, section, of platoon sergeant. All three are useful and develop different qualities.
Seek out opportunities to attend ABIC and BSNCO courses. Both will serve you well down the road.

One more thing from me to you that I have learned and preached for over two decades now: never stop learning. Our operational environment is dynamic and constantly evolving. You must as well. Too often, I have watched leaders and peers try to fight the proverbial "last war", while wondering why TTPs that were used in Haiti don't work in Bosnia, why what was used in Kosovo doesn't work in Iraq, why what was used in Afghanistan doesn't work in East Africa...
I'd like to think that we are smarter than that, but it keeps happening. Innovation is hard, but necessary in order to win while minimizing cost - something I've learned all too well when it is my resources being expended and my Soldiers getting hurt or killed executing.

Honestly, have a good picture make sure all your uniform candy is squared away. Your assignment history needs to look "clean", this means being consice and accurate and matching your NCOERs. Make sure you have and updated PT score and weapons qual REGARDLESS OF CMF/mos total soldier concept here. If you have college needs to be on there. If you have and OPI/DLpT/DLaP score with language get it on there. As for schools and ASIs NEVER SAY NO TO A SCHOOL!! Regardless of what it may be, and ENSURE that when our have a school you annotate it on your NCOEr by performing those duties from the school you attended. Seek the hard jobs and you will be rewarded. That's my best advice and I just got my sequence number for MSG on my second look in a VERY competitive MOS with only 13 years of service. I hope this motivates you. Good luck.