IMO, I don't think KM ever did arrive at 4JX the morning of 5/24, although I think his phone was there, having been delivered the previous evening when FD put the Suburban in KM's hands when FD left his dinner party to get meat after KM's phone polled the tower near 4JC twice. I think KM's phone was disconnected from the network when he passed it off to FD to give to MT, who - as the designated "cellphone manager" for the next morning - reconnected it to the network from roughly 7:30 to around 8:20 to give the appearance KM was onsite at 4JC.
When questioned about being at 4JC the morning of 5/24, KM seems befuddled. The time he gives doesn't match the time his phone indicates it was onsite. He doesn't seem to know anything about the AT call at 8:24 or 8:26 (I think there is a discrepancy about that time in the paperwork). And he denies seeing FD that morning, although it seems the whole point in having FD's phone at 4JC is to make it appear that FD is onsite for a meeting with KM.
I think KM was the designated "vehicle manager" for the plot. KM likely got possession of the Suburban the night of 5/23, gave his phone at that time to FD (who passed it on to MT), then he headed to a hotel near Middle Rd, where LE looked for cellphone activity between that location and 4JC around 10PM (i.e. after FD's dinner guests left).
I think KM's job was to get FD's Suburban (wearing altered plates) into New Canaan where it would be ready for body handoff after the attack and where FD would then take control of the Suburban, with KM then moving to the Toyota, to drive it (and the bike) back to 80 MS Rd. The Toyota arrived at 80 MS Rd at 12:22PM.
I have spent a lot of time with the vehicle and cellphone timelines, but I made a huge mistake and totally missed one important detail because I had artificially broken my timeline into parts (Morning, Afternoon, Evening) to make it manageable as I was working with it. But, then I combined the timelines into a single one and this important detail was painfully obvious: If the Suburban was in New Canaan the morning of 5/24, then whoever had control of it after the attack upon JFD had it for over 3 hours (roughly 10:30AM until 1:36PM) before it was next seen (as far as we know) entering the driveway at 80 MS Rd (FD driving) with MT arriving in the Jeep at the exact same time. It takes between 1-1/4 to 1-1/2 hours to drive from New Canaan to 80 MS Rd, so 3 hours of missing time for the Suburban allows for an hour and a half to do whatever was done in that time in which it appears at least at some point that FD in the Suburban met up with MT in the Jeep.
The possibility that FD and KM traded vehicles in New Canaan was something I had a small suspicion about because I thought I spotted something in one of the Merritt Parkway surveillance photos of the Toyota on its return trip to Farmington. It appeared (vaguely, detail is fuzzy) that the driver was wearing a wristwatch on his left arm as he held the visor by its left side to block his face. I searched every picture I could find of Fotis and rarely found him wearing a wristwatch and then finally, on one of the many videos of FD walking (near the courthouse, I think) I saw him with a wristwatch on his right wrist. So, does KM ever wear one? And if so, which wrist?
So, now we have three large unaccounted for spans of time: (1) 39 minutes after JFD's Suburban leaves her home driven by a suspect, but before the Toyota departs for Farmington (after accounting for driving time from 69 Welles to Lapham Rd and from Lapham Rd to the New Canaan Rest Area camera); (2) 1-1/2 hours after JFD's Suburban leaves her home driven by a suspect, but before FD's Suburban arrives at 80 MS Rd (after accounting for driving time from New Canaan to 80 MS Rd); and (3) roughly 40 minutes after FD's Jeep leaves 80 MS Rd for the first Avon, then Bristol, then back to 80 MS Rd trip from 2:24 to 3:55PM (after allowing for driving time). (1) is actually part of (2) if whoever was driving the Toyota back to Farmington assisted with the transfer of body and other things from JFD's Suburban into the other vehicles (FD's Suburban and the Toyota).