[EE] Small electromagnet cores?

Hey all-

I'm trying to make a flip-dot display like this one:
https://hackaday.com/2021/05/29/3d-printed-flip-dots/

I'm having a hard time with the electromagnet component of the project. OP
used 3mm threaded rod, and it seemed to work okay per his videos, but it's
not working for me at all.

I'm relatively sure the problem is in the permeability of the core.
Stainless steel is not a great magnetic field conductor. I'm looking for
something better.

Lots of searching has led me nowhere. I'm thinking of trying a nail next;
I'm wondering if anyone has any better ideas for a core? NB- I've seen the
"soft iron core" that Amazon sells; it's a bit expensive for my taste ($16
for 200mm; I would need four of those rods to complete the project).

Mike