How can I get recommendation letters for PhD application, while just starting a PhD program? I am starting graduate school (PhD program) soon, but there's a very high chance that I can't stay in this program for non-academic reasons. Now I want to apply for somewhere else, but I don't know how I can get reference letters. 

Since I applied for so many grad schools last year, my very patient and kind professors from my undergrad program submitted their letters for me at least 20 times. I bothered them a lot, they were very helpful, and I got into a relatively good program. I even went to them and gave them the good news of receiving and accepting the offer.  

Over the last few months though, my situation changed, and now there's a considerably high chance that I wouldn't be able to stay here after this year. I can't get any letters from my professors this semester, because I'm just starting the courses, and I suspect I won't even have the midterm results out by the time I'm applying to PhD programs for next year. I also don't have a an advisor yet until the end of the first year. So, it seems like my only option is getting the letters from my undergrad professors, but now I have two problems:

1 - I have their letters on a dossier service, so I thought about just using the service to submit their letters to those schools that accept the dossier service. But is that ok? Shouldn't I tell them that I'm using their letters? They submitted their letters for my applications last year, and they don't even know I'm applying again this year.

2 - Even if I could use the dossier service, most of the PhD positions I'm applying for, don't accept the letters from the dossier service. In fact, since I'm mostly applying for positions in Europe, I'll probably need to have the letters directly sent to my prospective advisors. How can I ask those professors for submitting their letters yet another round?! I can't even explain the reasons to them, as probably they won't want to read a long email with details about my personal life, and I'm not there to go and talk to them in person. Is it a good idea to have the dossier service email the letters to the prospective advisors?! Could I ask the schools/advisors to make an exception and accept the letters submitted through the dossier service?!

(I did my undergrad in the US, and I'm currently a graduate student in the US.)
Seems to me that your best course of action would be to send short emails to your current letter writers. You can thank them again for their letters, but state that, unfortunately, for [personal/medical/visa/whatever] reasons, you'll likely have to leave. (Note that there is no need for all the details! If they're curious they can ask you to elaborate, but the first message from you can indeed be short.) Then ask if you can re-use the letters from the dossier service for the positions that allow for that, or if they'd be willing to send letters directly. If they wrote their letters hoping you'd do well they'd likely agree to both, or, ideally, even to update and adapt the letters for the new positions.

I think that tactic has a better chance of working than convincing new professors to use a dossier service they're not familiar with, and it avoids your letter writers from feeling you went behind their back. Indeed, things could go sour if one of the advisors you apply with were to contact your letter writers to check if everything's above board and cleared with them.