My friend has this same need, and we’ve discussed but not yet executed…
Installing the hard drive as a second drive in the new computer, if possible, seems the best, cheapest, and fastest solution. If that doesn’t work, it gets more complicated.
Finding Win95 drivers for a network card might be a challenge, but Win95 was used for networking in its day. It even came with a browser!
While a Zip drive compatible with Win95 might be limited to 100MB and operate only over the parallel port, if you or someone you know has one, it would be a fallback option.
Because Win95 didn’t have USB support built in, and even Win98 (First Edition) didn’t have very reliable USB support, the “easy” option of using an external USB hard drive is not promising. But perhaps some of them might work.
I believe you could find a parallel-to-parallel solution (pcAnywhere?) to transfer the files, but that likely would involve an extra purchase.
Hope this helps.