RB -

You need a bit of elastic, with a soft plastic hook on both ends.

Isn't that what's called a bungee cord? (The other duct tape.)

Elastic (elastomeric) materials generally are hazardous to hardware, since nearly all of the common ones contain and exude sulphur and other corrosives. Short time uses in a situation like this might be ok, but the "emissions" spike rapidly when the rubber starts to fail, so a forgotton bit left too long where it shouldn't be can be quite harmful.

The paint department at your local hardware or lumberyard or (if you sneak in at night so your friends won't see you) even at Walmart has a variety of masking tapes, probably including the Scotch® low-tack variety that is intended for use where your masking tape must be left in place for more than the usual 24 hours. This stuff is specifically represented as leaving no adhesive residue on the surface when it's removed, and my limited use indicates that it pretty much works as represented. It's commonly available only in rather "bold" colors, so you're unlikely to find it in a color you'd want to display too prominently, and since it's a "low tack" it's holding strength is pretty low. (Left in place longer than usual on a wall it usually falls off under its own weight after a couple of weeks.)

If you're concerned about leaving a residue I'd recommend NOT USING ELECTRICAL TAPE of any kind. The adhesives on these tapes contain high levels of plasticizers so that adhesion is maintained when the wire flexes, and the residue that they leave when removed usually is "oily" and much more difficult to remove fully than even duct tape or masking tape residue.

Installing a "jack in button" with an internal jack so that you can swap pickups would be the preference if you expect to try out more than one or two different internal devices.

Untwisting (or twisting) the lead wire intelligently often can eliminate a persistent "loopiness" so that the lead "hangs where you want it" without glue and goop, at least long enough for an evaluation of a temporary device. Your office supply or craft store may have mini clips, clamps, and such that could be used for temporary holding. (Craft shops often have spring type clothespins as small as about 1 inch long? Your doll in the dollhouse has to be able to hang out the laundry.)

John