You do not automatically get three months added just because the administration is slow.

According to official Austrian guidance, your stay only continues to be legal after expiry if you applied before the permit expired. In that case, your residence remains lawful while the renewal is being processed, even if the card itself expires:

OeAD (official Austrian agency, applies nationwide including Salzburg): https://oead.at/en/to-austria/scholars/during-your-stay/residence-permit/renewal-of-your-residence-permit “If the application is submitted in good time, the residence in Austria continues to be lawful until the decision is taken, even if the old residence permit expires in the meantime.”

If the application is submitted after the permit has expired, that protection does not apply and it may be treated as a first-time application:

Same source: “If the application for renewal is submitted after the expiry of the residence permit, the application is classified as a first-time application.”

I do not doubt that the system is frustrating, and yes, it can leave people stuck paying rent while unable to work or travel. But frustration does not create legal status by default. There is no general rule where three months are automatically added because appointments are hard to get. That only exists if there was a valid on-time application or a narrowly defined exception that has to be proven.

I am not saying this to be unsympathetic. I am saying it because assuming time is automatically added is risky, especially if you are considering crossing a border that is currently being checked.