I just did this yesterday, connecting SV to GA in JED on the same ticket. Just go to the transfer counter and they will handle you. I had a sense the person there was basically waiting for me to show up, they probably flag these connections internally (as long as it is on the same ticket). Be aware - it’s a long trek. But yes, all airside without clearing immigration. I had a 1h40 connection, landed on time, and was boarding GA by the time I finally got to the correct gate in terminal H.
The process involves lots of x-rays, staff calling other staff, sitting alone on a transfer bus taking 25min, and waiting for people with the right authority / physical key to show up and let you through a door. But it seemed like all of the staff knew what had to happen, even though I’m sure they don’t get *many* pax making a connection like this. The staff isn’t that talkative either, don’t expect to get a full rundown of the entire process up front. You’re basically getting handed off from one person to the next until you’re there. “Sit here and wait for someone to show up” is a common instruction.
I chatted with the purser on board GA, and apparently I was the first transfer pax through JED she had ever come across

Tl;dr: for a T1 -> Hajj transfer, tell the transfer desk you’re connecting and follow their process, and you’ll be fine.
I can’t speak for the inverse direction, might be slightly problematic since it didn’t look like the Hajj terminal was built with transfers in mind.