The Svöðufoss waterfall may not have its own mountain, but it nonetheless is wonderful to photograph. I believe in recent years, you were able to get up close to it, however now there only seems to be a viewing point from afar. I’m assuming this because the area is a popular bird nesting area and there are many sheep that roam the area as well. So perhaps this is an effort to minimize human intervention. If you can time it right, sometimes the natural light during golden hour will shine directly on the falls for great effect.
Aside from old pop songs, chasing waterfalls is one of the most exciting things to do when in Iceland. As you can see, there is no shortage of them and no two look the same.
You can book day trips to see a few of the ones I have mentioned here, though the experience will likely be quite muted due to the sheer amount of tourists you share the attraction with. The best advice I have is to hire a campervan and drive around the country yourself, giving you the ultimate freedom to see these places at a time of your choosing, not to mention seeing some of the lesser known falls.