Overnight layover in Madrid

I'd suggest a hotel near the airport for the night of July 2. There are lots of them in Barajas with shuttle service. You could leave your luggage there and check in late in the day, giving you time for a partial day in Madrid. Madrid's Metro connects the airport to many points in the city. You'll need to get tested that day or the next morning before you fly. You should be at the airport about three hours before your US-bound flight, i.e. 8:15 AM, which is why I suggest a nearby hotel.

The airport website doesn't seem to mention any testing at the airport itself, but it would be worth inquiring further about that, or at least searching the site more thoroughly than I did. Your hotel should also have suggestions. Certainly there will be test sites in Madrid, which other posters may know about. Google can help you with this.

You have over 24 Hours in Madrid, arriving on a Saturday morning. I would get tested at the airport Saturday morning when you arrive in Madrid.
https://flycovidtestcenter.com/en/

Then take a taxi to the Europa Hotel near Puerta del Sol. This puts you in the center of Madrid. Check in (they have luggage storage if no room is ready), then walk down to the Mercado San Miguel for lunch. Explore the city. Why waste a full day by the airport. Sunday morning, take a taxi to the airport.

I'd stay in the city. Yea, it means you'll have to wake up a little earlier and spend a little more on a cab, or a little earlier still for a train/subway (which would be cheaper than a taxi from near the airport. But all totally worth it for a casual day in the city. It's not like you have to be well-rested to get on the plane.

Your arrival from Palma will be domestic, so you'll probably be "landside" before 10am and in the city center by 10:45.

Three hours is plenty (maybe more than enough) time for MAD. So, in a cab, I'd leave the city center at 7:45/8. Public transit, providing you plan a hotel convenient to a station, would be a 7:30-ish departure.

(Glad to offer tips on what stations- metro or train- to stay near. Answers might vary depending on the airport terminals you'd come from/go to. Which depends on the airline used.)

Just got back from Madrid and I second the recommendation of Hotel Europa. A taxi from the airport is 30 euros and will drop you off at one end of Puerto del Sol, with a short walk across the large plaza to your hotel. You can pick up a taxi to the airport on Puerto del Sol the next morning (they line up near McDonalds). I’m glad we were at the airport three hours before our flight because there are multiple checkpoints to go through (airline covid test check/security/immigration) and lines were long. But then I hate feeling stressed out and like to have plenty of time to catch my flight.

Hotel Europa will have you perfectly positioned for Rick Steves’ walk through the historic center of Madrid or you can hop the metro (only steps away) to other tourist destinations. You can do plenty in one day in Madrid.

I also recommend Hostal Acapulco, another of Rick’s recommended hotels on the 4th floor of a building a few blocks from Hotel Europa. We stayed there the night before our flight and taxis can drop you off right in front of the building and they’re lined up in the morning so hailing a cab was super easy. It’s a bit more basic than Hotel Europa but the beds were comfy, the bathroom clean, the WiFi strong, and they have complimentary coffee and tea in the lobby.

Thank you for excellent info. I’ve booked at Hotel Europa. I arrive on Air Europa from Palma, Mallorca. Besides a taxi what’s a little less expensive way to the hotel? I will have 1 carry on size rolling suitcase and small backpack.

anilytle- You'll arrive somewhere in terminal 1,2,3. From there to Sol, for public transit, I'd recommend taking the Metro from the Airport T1/2/3 station: Line 8 (pink) to the end at Nuevos Minisertios, LIne 10 (dark blue) to Tribulan, then Line 1 (light blue) to Sol.