All of the Gibson archtop offshoots I've played (Kalamazoo, Cromwell, Recording King, etc...) have fairly substantial necks. I think it's more common for necks without truss rods to be chunky.
In that price and age range it might be worth investigating Gretsch. There are Gibson L-48s and L-50s that might fit that bill too. The ones I've encountered vary pretty wildly, so I'd caution getting anything sight unseen.
Also possible one of the Chicago brands (Kay, Harmony, Silvertone) might have something that fits your hand well too, and should certainly be within that price range.
Epiphones are worth checking out too, my '41 has a very comfortable C shape.
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