[knc1]

You do want the mechanical support of the (weight of the) pins and any alligator clips.

You do not need the alligator clips, instead you can very tightly wrap the wire(s) on the pin(s).
Tight!!

My choice would be adhesive backed foam.
The sort of stuff that is sold as a gasket strip.

Get white if you can, avoid black and gray. Those colors probably come from conductive additives - which you do not want, it would create a short circuit between the pins.

Alternatives:
The 'rubber' stick-ons sold as non-scratch leg end feet.
Brown is probably alright, avoid black.

Any similar material + double sided tape to hold it in place.

Cut a small square of the adhesive backed foam, stick to the motherboard a small distance away from the connector, solder dots.
That is one less 'movable' piece to chase around while positioning the pins.

The pin, with the tightly wrapped wire lead on it, would go in the top of the foam and out the side of the foam, ending with the point at the solder dot.

Does the pin stay in place when you let go of everything?
Great!
Now push the pin point into the solder dot.
It doesn't have to go far, just enough to keep the pin from moving around or away from the dot.

That should do it for a good, temporary, connection.