I've only see a Small Form Factor use with Fiber connections and not for standard Ethernet CAT5/6 connections. I suppose you could just hard-wire your own cable to the board where the existing connector is placed.
I'm not sure if this is smaller than what is already on the board.
I wouldn't advise doing so - it's not quite as simple as that because the jack is an integrated magjack meaning that the Ethernet Hybrid Transformer is integrated into the jack. If you removed the jack, you'd need to connect an external transformer which would be quite a similar size (or be absolutely sure you have a jack with integrated magnetics) - if you don't know what it looks like, it's something like this: https://au.element14.com/pulse-electronics/hxu6200nl/ethernet-transformer-choke-350uh/dp/2911454
If you really need smaller, perhaps you should forego Ethernet entirely and consider using Wireless only if possible. That way, you might also consider a board such as the Raspberry Pi Zero W. Otherwise, you might consider a Model A variant with the USB port and use an external USB to Ethernet adapter. There is also a compute module for even smaller size but it is designed for embedded usage and has the minimal amount of peripherals on the SODIMM-form-factor card.
Interesting trivia was that the first Raspberry Pi Model B boards had no functioning Ethernet because they mistakenly fitted standard RJ45 style jacks rather than integrated magjacks - https://www.raspberrypi.org/blog/manufacturing-hiccup/
I didn't realize it wasn't a dumb port, like USB. Thanks for that info! Now I understand why it's so large.