Is there a conductive tape product that conducts from top to bottom?
What you want is foil tape with conductive adhesive - used for EMI shielding. Most foil tapes that I've found are not made this way, however 3M does have a range specifically for this purpose.

3M™ EMI Aluminum Foil Shielding Tape 1170, Conductive Acrylic Adhesive

You can find it through various suppliers:


Farnell (now Element 14 in
AU/NZ)
Amazon


Edit: You mention a sliding contact, so I thought it was worth mentioning that this product is just like aluminium foil - so may not be durable enough for that application. YMMV.