Probably cheaper to go to BKK and pick up a cheap ticket from there. Both TK and LH (not OW) have cheap BKK-SGN flights. Jetstar Pacific also have cheap domestic flights in the country. The distances in the country can be considerable, particularly from Hanoi to HCMC. Hoi An and Hue are lovely and worth a look if you get a chance, plus an opportunity to kick back and take it easy.

It's a really interesting country. Most (inc UK) visitors need a visa, there is the possibility of getting one through an agency in advance online, but not all such agencies are reputable, so you will want to do your research if going down that route.

NB if BKK does not appeal you can try transitting through HKG, where there are OW options (CX), plus there used to be cheaper flights with UA. But like the LH, TK flights above, they are, at most, once a day.