If you have changed your passport in Kosice, you need to report it to the foreign policy immediately. The best way to do it is to prepare a letter, print it, and sign. Then bring 2 copies to the Foreign police in Kosice (Trieda SNP 35, Košice, 04011) and let the police stamp one copy.
How to submit: go to room 220 (Stala služba) and tell, you wish to submit a paper. They will collect your passport and RP card and give you a confirmed paper (in case you brought 1 paper for submission and one for confirmation).
Another alternative is to send announcement by the Slovak post.