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Here is a (very)old post in reference to that: https://www.steelsoldiers.com/showthread.php?26374-1968-M35A2-W-W-tranny-problem&p=276453&viewfull=1#post276453
10 years later... we all have learned that all of the 3053 transmissions out there had one and the same 5th-OD gear ratio (0.79:1)...
Usually that is not an issue; the dogs from the synchros wear out faster.
There is a way to check for wear - on the work bench, at least: put the synchro in Neutral and stick the gear inside; then measure the distance left between tips of dogs, of both parts....
Another good test: slightly...
Checked your pics...
Input shaft is OK.
4th-5th sync has seen better days; specially its 5th gear end shows quite some wear on the drive side flanks of the dogs.
5th gear has seen some heat but may be reusable, after all. Rear thrust surface sure needs to be polished.
3rd gear: its...
Ah... you found the problem(!) - I mean the cause of it... That belongs to the 5th gear sleeve. Without the collar, the gear can move backwards enough to slip out of engagement.
I have found that collar split off of the sleeve, but always remaining in place; never broken in pieces like that...
I'm sure there are other pictures posted but here is one: https://www.steelsoldiers.com/showthread.php?23122-Deuce-trans-pushing-falling-out-of-5th&p=238242&viewfull=1#post238242
Maybe its time for an inspection and overhaul...
Synchronizers usually are the first suspects. But if for example the rear bearing (6308 ) on the output shaft already got some axial play, then things like you describe can also happen.
1st, try to grab the output flange and check for play...