Jeddah's heating up. While you can still find delightfully temperate evenings, generally it's already uncomfortable at night. Probably not a good opportunity to explore, especially with jetlag and carry on bags to deal with.

In any case your body will probably thank you for finding a hotel room. The in-terminal Aerotel (myaerotel.com) is super-convenient, and if you book six hours it's going to be around EUR100. Close to the airport, the Clarion is probably cheaper: it comes recommended by friends because of a free shuttle service to/from the airport: maybe they were lucky, I'd certainly budget for having to take an Uber in at least one direction.

You could always tough it out in the lounge. If you get there in time you can join the anti-socials and spread yourself out across seats/sofas and try for some shut-eye