308 Transfer case overflow tube

They did in some years put a hose on it and connect to the breather system.

As far as overflowing? Never once seen that unless it was overfilled.

What kind of oil is in it?

 

The only gear oil that a customer had me drain back out because he disliked it so much.

 

My ex-308 had that modification with the hose just run up into the upper bracket for the oil cooler with the end of the hose pointed down (had the extension tube and the breather valve was missing the top hat). It's not rocket science (but you may need to remove the "top hat" cap on the breather valve to have a good seal to the hose). Here's a shot of the much later system that Brian mentioned with the hose being connected to a nipple on the oil vapor recovery canister (and no separate breather valve on the bell housing -- just another nipple):

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The early 308 had the very short "breather valve" just mounted directly into the bell housing like this:

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While later on they added an extension tube (item 24 in this SPC figure) between the bell housing and the breather valve like this (don't have a picture of that handy):

https://www.ferrariparts.co.uk/diagram/ferrari/308-gtb-gts-usa/023-clutch-unit-and-cover

Do you presently have the extension tube or not? Is your "top hat" on the breather valve still in place or is it missing?

 

Well, if you've already got the extension tube that was Ferrari's "desperation fix" -- so looks like you need add the hose if you want to reduce the oil seepage (unless it's seeping where the extension tube enters the bell housing).