Some of the Lenovo tiny 1L machines can take a PCI card to add more ports. I've specifically seen it done with the 720q. You'll need a riser for it to fit. If you search on AliExpress for "lenovo thinkcentre tiny network riser card" or similar you'll find some examples.
There are several threads about this on Reddit, but I'm not going to link them because of all the mess going on over there. If you search a bit you'll find them though. If they've been deleted due to the protests, try a cached copy on Google.
I personally use a Qotom PC. Not sure if it fits your exact requirements since you can’t get a PCI card into it, but it does have 4x1Gb ports. Works really well with opnsense.
I've been eyeing one of these. Just haven't pulled the trigger. Been looking at these and other 'lesser known' brands. How do they handle heavy workloads? Any temperature issues being that they are passive?
I’ve only been running a few services like AdGuard Home so I can’t speak to very heavy loads. I’ve had no issues with it. I probably wouldn’t run my Plex server on it but any regular network services should be fine.
I have mine sitting on a wire metal shelf so it gets plenty of air flow. I did misconfigure the Unbound DNS server to recursively query itself and CPU load was at 100% for several hours. I noticed it was warmer physically but it was not hot enough to give me any concerns. Still well within what’s safe to run.
Serve the home recently did a write up on a HP Elite mini 600 G9. Apparently there is an adapter, HP Flex IO V2 module, that adds a 2.5gb port. Worth a look, but I would just get a SFF pc to have more pci slots and options.