Handshake has a job he wants you to take down in Gray Zone Warfare, which he wants you to do on the side. It's not an official report. For the True Grit quest, he wants you to hunt down the antique revolver that John Wayne used.
True Grit Quest Guide – Find the Revolver – Gray Zone Warfare
While it’s not an official job, it certainly pays. You’ll need to do everything you can to find this antique revolver and return it to Handshake in Base Camp. Tracking down the location of this revolver in Gray Zone Warfare will be a bit tricky, but Handshake does provide a general search area to make it easier to find.
I found the Antique Pistol by exploring the Motel at Hunter's Paradise, which is located east of my Base Camp. I completed this mission before Crimson Shield International, which means you may have a different location when you speak to Handshake. The general concept and location should be identical, however, and the Antique Pistol should spawn in the Motel, close to where you completed On The Range.
You can find the antique revolver in a specific room. Unfortunately, the room is blocked by a locked door, and when you get there, you need to find the Motel Key for 102. The beginning of the name of the key varies depending on the location you visit. For me, it was Hunter's Paradise. It's a similar process to the Helping Hand quest.