If you are putting the jumper cables on the battery then it is almost surely lack of output from your battery. It could be a weak battery or simply bad connection from the battery to the cables. Did you clean the cables when installing the new battery? If not then do so and put some dielectric grease on the posts. Also, the battery could be weak because the generator is not charging it properly. That could be caused by a bad generator or regulator or also bad contact with the battery cables again. Of course, it could be a bad battery. If you are connecting the ground jumper to a chassis ground, like the books tell us, then it could also be a bad ground connection somewhere, battery cable or engine/chassis cable.
Whatever you do, check the obvious things first. Don't go buy a new generator or solenoid until you know for sure that is the problem. From your stated symptoms, those are the lease likely cause anyway.