Seaway Export offers car export services from USA to Hungary via following ports of loading (POLs):

Port of Savannah

Since Hungary is a land-locked country, ocean containers can be delivered via land transportation from the ports of Ploce and Rijeka (Croatia). 
Service is also available with full containers or de-consolidated cargo being delivered by trucking from Croatia. 

The major shipping lines servicing these regions are Maersk, MSC, ZIM, CMA-CGM, Evergreen.

Seaway Export will deliver a full container under ocean Bill of Lading to the closest ocean port or agreed in-land location, but final delivery to consignee within Hungary will be responsibility of the consignee / importer.