Reapplying to a PhD program while attending another one
I was able to transfer from Georgia Tech to MIT after a year. I initially went to GA Tech to study with a famous professor, who, once I got there, said that she was slated to move to Carnegie Mellon in 2 months. Nobody else at GA Tech was doing what I was interested in at the time, so I decided to go through the application process again. I re-used one of my rec writers from my app to GA Tech, and got 2 new rec letters from GA Tech profs I had worked with. I was able to do this because of a great program GA Tech had at the time (I hope they still do), wherein first-years do 3 mini-projects with 3 profs in their first semester. 2 of those profs provided recs, one of which had come from the MIT department I ultimately got accepted to.

To do something like this, you'll want to do the same and hit the ground running at University Y, starting research projects immediately despite your first-year courses. This will give you recs, and also give the admissions committee at X something new to consider.

One tricky thing is if/how to present your intentions to the profs you work with at X. If they know you're planning to jump ship, they'll be less invested in you. It's probably best if you don't come in saying that you'd like to leave for X ASAP.