American Airlines, Finnair, and Iberia have round-trip flights from San Antonio (SAT) to Helsinki (HEL) for $598 with a connection. Flights October to March. Departures on Monday, Tuesday, Thursday, Saturday, and Sunday. Extended weekends available.
Some small OTAs on Skyscanner have this for about $577. If you plan on booking there, beware of any site that charges a fee to cancel a ticket within the 24-hour “free” period. Use unknown booking sites at your own risk.
A Carry-on bag and also a personal item is allowed for free on. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.