The wine sector is facing a major challenge in wine packaging. The environmental impact of packaging, one of the most CO2-emitting aspects of the wine industry, needs to be addressed. In this study, we investigate how open Hungarian consumers are to buying wine in non-traditional packaging. We showed images of several alternative packaging options to a sample of domestic consumers: 1) a domestic wine bag-in-box and 2) a pouch package and (3 & 4) their internationalized equivalents. We then presented participants with a questionnaire. The questionnaire asked about the following topics: Traditionality, Familiarity, Purchase Intention (PI), Product Appeal, Taste Expectations, and Uniqueness, with participant demographic questions also included. Cluster analysis was conducted for both Hungarian packaging (HP) and International wine packaging (IP) options, suggesting that different consumer groups have different attitudes toward alternative wine packaging options. Overall, consumers are slightly more accepting of Hungarian-packaged wines on average, but at the cluster level, if only those consumers most likely to accept (and purchase) alternative packaging are considered, acceptance is greater for IP wines. Generation Z wine consumers, who are self-reported to be knowledgeable about wine and have above-average incomes, were the most likely to accept international and Hungarian alternative packaging options. The innovators’ cluster group demonstrated the highest intention/willingness to purchase wine in alternative packaging of either the Hungarian or international sort. The mean value of Innovator IP is 4.84 for bag-in-box and 5.01 for pouches, making this group more accepting of pouches than bag-in-box packaging—an inversion of the preference of most other groups investigated. For both bag-in-box and pouch, the mean for the Innovator HP cluster is 5.17. For both Hungarian and international packaging options, it is clear that acceptance is lower for pouches (PI: 3.27; 3.47) than for bag-in-box (PI: 3.62; 3.88). We offer our results to decision-makers so that they may better understand the preferences of the Hungarian wine market and work to improve acceptance of non-traditional packaging options through consumer education and familiarization.