Icelandair has a policy called the “Iceland Stopover” that lets you break a transatlantic flight in Reykjavík for up to seven nights at no extra fare. If you are flying between New York (or Boston, or Seattle) and most European cities, you can add three days in Iceland and it does not count as a separate ticket. It counts as a stopover.
I use this twice a year. It turns a flight into a mini-trip. Reykjavík in three days is not enough time to see Iceland — but it is enough time to remind yourself that a city exists at human pace, walk to the ocean at sunset, eat one really good lamb dinner, soak in a geothermal pool, and get on the next flight refreshed instead of destroyed. The cognitive difference between arriving somewhere after 8 hours in the air and arriving after 5 hours plus a three-day break is enormous.
The catch: you have to book with Icelandair as your primary carrier, and the stopover has to be either at the start or the end of the trip, not in the middle. Their website makes it slightly hard to find; the option is on the checkout page under “extras.” If you fly the Atlantic more than twice a year, this policy alone justifies choosing Icelandair over the big US carriers. It has changed how I plan.