Skipping in the smaller cogs may not indicate a worn cassette or freewheel. Those cogs usually get the least wear for most cyclists.
Some of the most likely reasons for skipping:
- Mismatch between the quick link and chain. Check the pin lengths. You may need a fairly accurate gauge for this to distinguish between 7.1mm and 7.3mm.
- Mismatch between chain and freewheel -- I'm guessing freewheel in this case since you mentioned 7-speed.
In the first case, any binding with a mismatched chain and quick link can cause skipping.
In the second case, some freewheels are picky about chains. In particular the Shimano MegaRange freewheels want narrow chains. The spacers between cogs on MegaRange freewheels are a little wider in diameter than some other freewheels. A new chain that's too wide, by a tiny fraction, will contact the spacers in the smallest cogs, causing it to climb out of the teeth and skip under pressure.
I haven't seen the second problem with Shimano 7-speed cassettes, only with their MegaRange freewheels, the kind with the big jump to a 32 tooth or larger granny gear. I haven't experienced it with SunRace freewheels or Shimano Tourney freewheels, but that may be because of a correct match between chain and freewheels.
So if your bike has a Shimano freewheel, be sure to use a chain marked "narrow" on the link plates. And the correct quick link.