First, my screen caps.
I've often wondered if this hat was something other than a Herbert Johnson but haven't given much time into research on it so I thought I'd post here to get some opinions.
I figure it most likely an HJ like the rest, just bashed differently. We see Ford bashing his own hat and messing with it several different times but that was his main fedora. The fedora I'm talking about only seems to have been used for the scenes right outside the Temple entrance, both before and after Indy goes in. Once Indy is inside it's a different fodora. I also thought that if it was used for all the outdoor Kauai scenes then it may be that after it got completely soaked in salt water it was discarded and a new hat used.
I also recall a thread where someone said that another hat maker made some hats for Raiders in a Pecan color, had seen pics of them being made as well as the receipts from Paramount. So, we know other hats were made for the film but we have no idea if they were used and if so, for what characters. I thought that it may be possible that the Temple exterior hat was a different hat than the others in the films. Maybe they intended it to be that way, as if his hat was ruined at the beginning of the film and he simply grabs another from his closet.
The hat does appear to be slightly lighter, or more Pecan in color to me than the others in the film, but that opens up a whole completely different discussion which I know leads to nowhere fast, but I thought I'd throw that out there.
The front pinch looks tight but the sides are not pushed back which is evident in all the other hats in the film, yet it retains it's stovepipe shape. Now, I can't do that with my hat. If I don't push the sides in far enough I get a slight taper. Of course, it's not blocked on the same block as the original either. I was thinking that maybe this is the hat that Swales blocked in the pinch with that machine? Any thoughts?
From the pics the channel seem to be a fair depth although you can see the back is not pushed as far down as the other fedoras in the film. You can't really tell if there's a difference in the depth of the channel in this hat to the other hats but the crown looks alot taller on the Temple hat and the ribbon a little smaller. However, the hat experts here can look at the bow and see some things I can't.
The brim looks alot different too and that's very evident in the pics. Alot flatter.
It's most likley just an HJ...perhaps the very first prototype that they came up with before placing the order for the rest? I have that theory about one of the Raiders jackets which is the main jacket worn by Ford throughout the film.
Any thoughts about this hat? I know theres been some sort of discussion about this hat before but the stupid search won't let me get past one page of searches so I hope I'm not being redundant.
). So - especially with very soft felt - IF this hat was made on the same blocks as the other hats, the deep top crease in the rear IS the reason to the reverse taper... But back on topic.
When Noodleman mentioned the hat had a tall crown, we must remember that starting in the late 50's, or early 60's, crown heights dropped overall. So, from that standpoint, it was a tall crowned hat. When I talked to Stetson around 15 years ago, I was told that the tallest dress hat that they made, or had blocks for was 5 inches max. If the original Raiders HJ was 5 1/2 open crown, then that would be a tall hat. 1/2 an inch on a hat shows up as a mile. I have had the pleasure of reblocking many HJs over the years. Most were from the late 80's and 90's. All of those hats had a 5 1/2 open crown, with the exception of two. I saw one with a 6 inch open crown, but came new with a very tapered blockshape, non Indy. The other one was 5 inches tall, and was a squareish block, and looked too short for an Indy fedora. I also heard of two more accounts where Swales sent short crowned hats. I think he just sent what he had on the shelf, at least at one point in time. 
I saw that too. Just going by the width of it in comparison to the width of the ribbon, it looks just like the bottom side of the reeding tape. But, the thing is, I have never seen a hat that would show the outline of it, except a very, very thin vintage Borsalino that I owned a few years ago. A roll up. It was so thin and soft, that the reeding tape would show, prior to attaching the ribbon. I guess if my ribbon would have been thin, just like the fabric used on the Akubras, it may have shown through as well. Something to ponder. Now, before a vintage hat guy chimes in with, "Borsalino did not use reeded leathers on their roll ups", let me explain. I removed the non reeded, whipped in stock sweat, sized the hat up one size, and then had to install a reeded sweat. You can't sew that other sort back in, without a special machine. 
If what we see is indeed the bottom portion of a reeded sweat, two things must be present. First, the felt in that part of the hat is thinner than most. And, it has to be really soft. Now, you could say, " but look at the thickness of the brim. That hat was not thin!" I would counter with a little known fact. Back many years ago, when there were many small custom hat shops competing with the likes of Stetson, Knox, Lee, Cavanagh, Dobbs, etc, some of those hat shops were run by master hatters. One thing that a master hatter would do when making a hat was vary the thickness of the felt. The tip, or top of the crown was the thinnest, and as it dove down toward the brim, the felt would get slightly thicker, finally ending up at the thickest part-the brim. Some of this was done by the way that it was blocked, but part of it was in the pouncing too. So, a finely made hat was the thinnest at the tip, and gradually became thicker as it reached the brim. Of course, the brim was left thicker, and was the same uniform thickness. So the thinner felt stopped at the brim break. If this Raiders fedora was made in this way, that would explain what you see outlined through the ribbon. It is possible. Fedora
Fedora
Come to think of it, I learn something new just about every day on the net. Alot of it here from these talented people.