If you have a hammock, you should take it to Grefsenkollen utkikkspunkt and hang out there for a day (or night). It is quite popular, so you might meet others there, but I have also had the place to myself for hours.
The viewpoint gives you a wide view of Oslo, but you are much closer to the city than at Grefsenkollen and Holmenkollen. You can hear the city below and study life as it is happening, while also enjoying bird song, a view of the fjord and the green surroundings. There is a bench there, and many trees that are perfect for hanging your hammock. Make a stop here on the way to/from Grefsenkollen, or bring some food, drinks, a book, and your hammock and make a day of it. I have never spent the night here, but if you are up for it I imagine the views of Oslo by night are magical.
The viewpoint is marked on the map, and I recommend using Google Maps to find it. There are a variety of small paths leading to it, either from the road or the hiking path to Grefsenkollen.