1989 Toyota Corolla having trouble installing passenger-side drive axle

1989 Toyota Corolla mileage: 125,500. I found a little trick to help me pry out a stubborn passenger-side drive axle from the 3 speed automatic transmission of my 89 Corolla. Unfortunately, I'm now having trouble installing the new one. The axle makes it past the seal and I can feel it contact the gear inside the transmission. While holding the drive axle in place, I slowly rotate it while maintaining a firm and constant pressure until I feel the spline on the axle match up with the gear in the transmission. At this point, I try to push the shaft in, but it just won't go. I can see no foreign objects in the pathway. I cannot see any damaged splines. There is no key way forcing the shaft to go in only one way. I don't want to damage any bearings or splines etc, so I'm reluctant to try the hammer. However, I don't have the body strength to just push the axle back in place. Could you give me any hints as to how to install the new drive axle in my Corolla?