Transmission Cooler Fittings (Question)

[MagmaJct]

Background: PO installed an external transmission cooler mounted several inches in front of the radiator. The lines were removed from the radiator and re-routed up the front of the radiator where 3/8" fuel hose was slid over the inverted flares leaving the nuts intact.

Problem: When towing I get the smell of burned rubber. I suspect the external is undersized, and being so far from the radiator, gets little air flow from the fan.

Planned solution: Re integrate the radiator cooler, and run the external cooler in series.

Situation: External cooler has 3/8" hose barbs. I'm trying to find the correct 5/16" inverted flare to 3/8" hose barbs, both for the radiator, and the original metal line. I ordered Jegs 60340 which is the wrong thread.

I've found Hayden 393 from many sources on the internet. But I can't, for the life of me, find the hose barb size. I need it to be 3/8".

Does anyone know for sure?