I’d look on hornet’s nest…I got a lot of good info from the SV650 forum, ended up with the right springs/oil/etc in the front and a ZX6R rear shock, but that is very specific for my bike/weight so I think you’ll need to do more research.
Yes the Hornet 600 does bounce around a lot on the limited std suspension. Winding up the preload to it’s max is about all all you can do on the rear.
An aftermarket rear shock is a good idea and anything should be better than the std kit, check out the websites that have been suggested and you might be able to pick one up at a sensible price. I’ll check out what price we can get a Hagon for and will let you know.
The biggest problem I’ve seen on the Hornet 600’s is with the front end, it’s a pogo stick. Transform it with an easy cheap fix: linear springs and a slightly heavier oil or even cheaper just change fork oil with (from memory, will check) a 15w.
The standard Hornet shock is very difficult to rebuild, some Honda included say impossible, some firms claim to be able to do it. But whats the point? it ends up costing loads. And you still end up with a soggy a**e. financially you are better off fitting a Hagon. Which will give you a nice firm ride.
Forks good quality (linear) springs of your choice, thicker oil, personally I’ve got 20w but I like a tight front end. Also a longer spacer - by about 3/4" works well. Having the oil holes drilled out a bit is another mod worth doing. BSD in Peterborough did my forks. Transformed the ride.
Have you set the static sag on the bike ?
Preload is basically putting tension on the spring which, contrary to popular belief doesn’t ‘harden’ anything up, but it does raise the ride height.
If you’e going to go down the rebuild or new shock route, it’s worth getting a spring matched to your weight.