1. Go to the health insurance provider (OEP), and report that you worked abroad. This is necessary because everyone is required to have insurance somewhere. If they are not informed about this, they only notice that no one has been paying the Tb for a while. Therefore, it may happen that after a while you receive an invoice with the amount of contributions, interest and penalties. You can authorize the health insurer to collect the fees on your behalf. In that case, however, it may be necessary to wait months for the case to be closed.
2. It is worth going to the employment center and inquiring about the conditions for claiming unemployment benefits. If there are any, bring them with you. U1-es form, if in this case they cannot be trusted to request them in person.
3. If you have an annual job and went on vacation only to return home, so your main place of residence is in Austria, where you have your permanent address = Hauptwohnsitz, then you must register from Hungary. Go to the local municipal office and you will receive a new address card with the “foreign address” label. You must surrender your ID card; now you can only identify yourself and cross the borders with a passport, so if you don’t have a passport here, you can also apply for one.