COS From H-4 to F-1 (I-20 Update)

I have a quick question and would like to give you a little background information before I ask.
I’m currently on H-4 visa (expiring Sept 2021) and had applied for my Change of Status to F-1 using I-539 at USCIS on May 3rd. My Master degree course started this June and will end on June of 2021. I used my I-20 with June as the course starting date. My application is on pending stage and I’m in my second quarter of the course now.

I had requested my college to issue a new I-20 so that my SEVIS does not get auto-cancelled. And they issued me a new I-20 with Jan 4’th (3rd quarter) as my new starting date.

I would like to know if I can upload a new I-20 at USCIS with January 4th as the course staring date without having uploaded an I-20 for my second quarter (September as the starting date)? Would it create any complication?

Also, Just wanted to know if I have to update any other details along with I-20 in my I-539 application?

(As per the case processing times, I can expect my application to be processed by mid of November.)

Hoping to hear from you soon.

Thanks & Regards,