German health insurance (Trier)

First, I apology for maybe almost the same question like the other ones from these theme.
I would be really glad if you could give me some directions or advices about insurance here. Currently I am aupair and I finish it in September. After it I want to stay here and find a full-time job. Before that time I want to do mini-job but of course nobody does not want to employ me with no insurance (aupair one is not valid for other jobs).
As I know I can not ask for public insurance if I am not employed, so I was thinking about taking private one untill I find a job and then change to public one.
Could I have some difficulties with it ?
Thanks a lot for any tips!