score:2
I suspect your cheapest option would be to find a round-trip that goes to Vietnam via Taiwan, which will be priced competitively with other US-Vietnam flights, then add a stopover in Taiwan on way over, which would typically cost around $100 extra (YMMV).
Your two options for this would be Taiwanese carrier China Airlines and EVA Air, or if you flip the itinerary and stop over in Vietnam, Vietnam Airlines.
As tickets like this are difficult to impossible to book online, I would recommend exploring your options with ITA Matrix and then using a bricks'n'mortar travel agent to do the actual booking.
Upvote:0
There are many options to explore.
You can look at round trips from Houston to both Vietnam and Taiwan and the costs for round trip between Vietnam & Taiwan. And you need to look at these flights separately, as the inter-Asia prices can vary depending on your starting gateway.
You can price out an open jaw ticket (fly to one city and return from another), then look at the cost of a one way Inter-asia ticket to connect the two. Again you need to look at both directions.
Finally you can book a multi-city itinerary going from Houston to Vietnam to Taiwan to Houston (or vice versa).
While the rule of thumb is that a single round trip is cheaper, that is not always the case. You can find excellent open jaw and multi-city deals, it all depends on the dates and what seats are available on the flights.
Upvote:1
Use the multi city feature on your preferred booking engine to see if you can construct an itinerary stopping in several cities at a reasonable price.