@madjon is on the money mate. Without the a thermal layer beneath you (thermal mat of some decription then yes you will wake to the cold. The only time I dont freeze when using my hammock is in the tropics.
However if you go to a synthetic/quilted bag then this will reduce the cold somewhat but the trade off is size/weight.
The thermal mat does not need to be huge or flash. If bulk is an issue then concentrate on just covering your core.
Bookmarks