KMC Z8S IG or HG compatibility
That model number you provided does not end in the numeral "1". Rather, it is the letter "I" which stands for "interactive glide." That means that although it is of the Shimano HG model series it is only compatible with IG chain. IG chain has an external width of 7.1mm compared to HG's 6.6mm.

Your KMC Z8S chain has a outside width of 7.1mm. It is therefore in the family of Shimano IG compatible chains along with KMC's Z51 and Z72. What you may try is adjusting your B tension screw on your rear derailleur. Adjust it so that your upper carriage/gear pulley is close to the low gear sprocket but not hitting it. Doing so will improve chain to tooth engagement. 

Also, check that your derailleur/gear hanger is not bent and that your derailleur limit screws are properly set. 

If this doesn't work then your sprockets may be worn despite appearances. I have encountered freewheels/cassettes that appeared un-worn but exhibited the slipping you mention even after installing a new chain. The problem was solved upon installation of a new cassette. Slipping sensation may also be caused by a worn or damaged hub body (aka, freehub) as the pawls fail to properly engage the freehub teeth. In such case it isn't the chain but the freehub itslef that is slipping but it can feel like a slipping chain.

Lastly, if your chainline is way off you may have chain rub or slipping front or rear. If you recently changed the bottom bracket, serviced your rear hub, or swapped sprockets within your cassette cluster check that all spacers were replaced properly. Also verify your new bottom bracket spline width and offsets match the old.