The following is offered FWIW because we just got super lucky.
Our plan was to start looking for a qualifying lease for the D7 beginning 1 September. Following our scouting trip in May/June, we developed a list of neighborhoods and a list of criteria for an apartment. We planned to spend September doing it ourselves and sticking just to apartments that fit all criteria. The October and November plan (assuming a D7 application target date of 1 December) was to 1) get pros involved and 2) start giving up certain apartment criteria until we got a lease.
So in late August I got an e-mal alert from Idealista about apartment in our first-choice neighborhood and in a building I had admired during scouting. Plus, it was a pro advertiser (development/real estate company) -- another thing we focused on. I saw no reason to wait until September, so we went for it.
Lola and I decided we'd initially double-team the advertiser. She would call, and I would employ a combination of WhatsApp texting and using the contact form in the ad. Side note: It's important to use WhatsApp because its ubiquitous in Europe. We got up at 3 a.m. (we live in the central time U.S. zone) to start this process and were both in contact with the agent immediately. After a bit of communicating, the agent realized she was being double-teamed (she was chill about it) so Lola passed off all comms to me. I spent the next week getting up at 3:00 to text with the agent through WhatsApp.
We were lucky to know some American expats in Aveiro. We arranged for them to visit the property. After a good report, it was a day until we sent a deposit and signed a reservation form. Then just a couple of days until we received the lease. 24 hours later, a lawyer we employed in Aveiro looked it over and made a couple of minor changes.
Two days later everything was signed. The place is ours 🙂
We literally got the first place we tried to get. That's just extraordinary good luck. So I'm not sure if there's much to learn from this. Thus FWIW.