I grew up on a boat with no refrigeration, and we kept open jars of jams for several months. As long as you don't contaminate it (don't use the same knife for multiple condiments!), it's generally fine.

Same thing with jars of Mayonnaise - that one gets you strange looks.

Homemade jam, and some commercial jams will keep at room temperature for a looooong time. The main form of spoilage is mold growth, so try to keep the lid on as much as possible. Close the lid back before you spread the jam on your bread. It only takes a couple spores falling in to start it going moldy.

As I understand it the main thing is a lot of jams not having enough sugar in them anymore to prevent spoiling.

Jams and jellies existed before refrigeration and were used to preserve fruit year round. Root cellars kept things cool and you would just inspect your food before eating. Pickles is another thing. They used to be stored in a big barrel and people would reach in and grab a room temp pickle, safe and preserved. Today, they're always fridge after opening. It's safer, but it's a spectrum of risk, not binary.