Suppose it is October 1st, I am a first-year math Ph.D. student at University X (medium rank university), and suppose I managed to get good reference letters in the summer/fall thanks to ongoing collaboration on an article that attracted the interest of well-known professors of top rank universities (with whom I am in contact and that I did not know one year ago at the time I was applying for my Ph.D. program). Is it possible to apply again to more prestigious universities even if I have already started the program? I would of course terminate the Ph.D. I am already in: would this cause any bureaucratic trouble in the University I am already in?
I can't say if this is universally true, but in the US and, I'd guess, many other places you can do as you like. Of course you need to be accepted by another program and the standard may be higher at a new place, but you can apply.
Most places you will need to honestly divulge that you are already in a program, but the opportunity to work with some specific professor elsewhere can be a valid reason for changing. Increasing your opportunities is a valid path.
You might explore the likelihood of acceptance before you apply, given that you have contacts to do so.
It isn't even necessary to inform the current place that you want to move until your leaving would interfere with ongoing work and affect others. One year into a program isn't very far, actually. And, in math, disruption of others isn't very likely.