Great trip report!
A few things to add...
1. SATA provides free inter-island flights to the other islands. You don't get to choose which flight, and it's usually the first non-stop to that island. If you time things well, you can spend 24 hours at your gateway, and your inbound and outbound gateways can be different (e.g. fly into PDL, out from TER). So theoretically you can visit 4 islands for the price of 1, spending 24 hours at your inbound and outbound gateways. There's an inter-island ferry between Horta, Pico and São Jorge, so you can even do 5 islands for the price of 1 if you time it well

2. I'll +1 Ilha Verde - not super cheap (they're usually cheaper through consolidators like economycarrentals) but they do have new cars.

3. The highest point in Portugal is the top of Pico Island. It's a must-do hike for hike-lovers. It's not an easy hike - you go up about 1200m over only 3-4km over a very uneven terrain. The views are worth it though.

4. Lots of hiking on São Jorge.