It's unlikely to be the derailleur tension, as the drive tension pulling from the crank far outweighs what the derailleur provides. It's really only on lumpy off-road stuff that low derailleur spring tension can cause "ghost shifts" as it flaps the chain around.
Check for a stiff link where you joined the chain (assuming you used a pin rather than a quick link). A stiff link will not mesh properly, especially on the tighter-radiused smaller sprockets, and you'll get a slip each time the link hits the derailleur & gears. It's often not as apparent on the bigger sprockets because as well as the bigger radius resulting in less link movement, there is greater chain wrap to spread the load, so it's less likely to jump.
Easy test: drop the chain onto the smallest sprocket (choice of chainring doesn't matter a lot) and backpedal. You'll soon see if there's a sticky link 'cos it'll catch & flick the derailleur as it passes through.