Hyundai has risked confusion by calling its new model the Santa Fe — not to be confused with its five-passenger Santa Fe Sport, a similarly styled CUV introduced last year. Hyundai decided to drop the Veracruz tag used on its outgoing seven-passenger CUV, a name that surely would have resonated better in Mexico.

The new Santa Fe is the latest salvo from the South Korean automaker, whose presence in the U.S. continues to strengthen by dint of new vehicle models that, by all accounts, are being received warmly by American consumers. Hyundai has helped its cause by maintaining production discipline and avoiding the cycle of inventory gluts and deep price discounts that have hurt other manufacturers.

“Our chairman has been very clear that we’re going to concentrate on quality, not on opening new factories as fast as possible,” said John Krafcik, chief executive of Hyundai’s U.S. arm. Chung Mong-koo is chairman and chief executive of Hyundai Motors Co. and its affiliate, Kia Motors Corp. Speaking about Santa Fe and other Hyundai models, Krafcik said the company’s U.S. sales have been constrained only by the ability of factories to meet demand. According to Edmunds.com, Hyundai dealers had 42 days supply of vehicles in inventory, compared with 64 days supply on average for the industry at large.

MORE: ZipCar CEO resigns, hours after Avis acquisition

Santa Fe is positioned to compete primarily against Honda’s (HMC) Pilot, Toyota’s (TM) Highlander and Nissan’s (NSANY) Pathfinder, and secondarily against the Ford (F) Explorer and Mazda CX-9. All the models are roughly the same size and have three rows of seats, which make them attractive for families.

The Santa Fe Sport, the variant introduced last year, seats five in two rows, plus rear cargo space. It competes against the Ford Edge, Chevrolet Equinox, Nissan Murano and others. Crossovers have grown to about 21% of the U.S. vehicle market by of the end of 2012, from 13% in 2007. Traditional SUVs, by contrast, have fallen over the same period to about 4% from 10%.

Crossovers — what were called “station wagons” in antiquity — are inherently more fuel efficient due to unibody construction, which is similar to that of a passenger car. SUVs, by contrast, are heavy, powerful, and relatively fuel inefficient because most were built on truck frames.

MORE: Can General Motors remake its rental car?

Alec Guitierrez, a senior analyst at Kelly Blue Book, said Hyundai’s decision to drop the Veracruz name most likely stemmed from slow sales of that model. With only 12,700 units sold in 2007 (and 8,232 all last year), the Veracruz hardly put a dent in the sales of other models with three rows, such as the Buick Enclave and GMC Acadia, both built by GM (GM). The Santa Fe, by contrast, contended well against the Murano, Mazda CX-7 and Toyota Venza. “Santa Fe has considerable brand equity, and consumers have grown to trust the nameplate,” Guitierrez said.

The new model will round out Hyundai’s three-pronged approach to CUVs: the Tucson for “pre-families” — a.k.a. singles and couples — Santa Fe for families, and Santa Fe Sport for “post families,” those whose children have flown the nest.

For now Chairman Chung’s preference for quality over quantity is paying off for Hyundai and its dealers in higher transaction prices on its vehicles. The temptation will be strong in the not-too-distant future to expand its U.S. production network, perhaps by building another assembly plant.