How do I make the chicken absorb more salt by freezing it?
I think your best bet is to change your recipe. While trying to figure out the exact process that sometimes makes your soup work does indeed sound like an interesting science project, it doesn't sound like its going to yield a reliable recipe. Especially since you've got to deal with supermarket chicken from suppliers that may change their processing procedures whenever it suites them—maybe even from package to package, depending on which plant it came from, or the specifics of the chickens the plant processed that day.

There are easy, reliable ways to get salt into chicken. The following two will get you salty chicken, every time you do it:


Put your chicken (chopped up or whole) in the fridge submerged in a 13% brine for a day, and you'd at that point have chicken which would be (once cooked) inedibly salty (among other problems). 
You could chop your chicken fairly thin, and pack it (again in the fridge) in kosher salt. Then it'd become dry, and also very salty.


Of course, that'll be far more salty than you want. So you'll want to scale back—use a 5–6% brine, put it in only for a few hours, etc. But that will get flavorful chicken every time.

Then, to keep your soup base from being salty:


rinse the brined or salted chicken before adding to the soup (to remove any salt resting on the surface)
keep salting of the soup to a minimum.
don't overcook the chicken, that'll force more liquid from it.
make sure to use low-sodium chicken broth. Normal store-bought broth/stock is pretty salty.
if too much salt leaches from the chicken, cook the chicken separately and drain it.


&lt;/rant>

edit: random suggestions since the above apparently doesn't work


Commercially, many things are quick-frozen (e.g., fish) to prevent ice crystal damage. It would seem to follow then, that since you're trying to cause ice crystal damage, you want to freeze as slowly as possible. Easy way to do this would be (assuming your chicken is already under 40°F e.g., in the fridge) to insulate it before throwing it in the freezer. So, put it in a freezer bag, but then wrap the freezer bag in some kitchen towels, then toss that in the freezer.
In previously-frozen (commercially) chicken, there may be some anti-ice-crystal additives, I have no idea. Previously-frozen isn't always sold frozen. Check the package, it should say (probably in tiny print).
You could try a second thaw/freeze cycle (just make sure to thaw in the fridge, or in cold water, not the microwave, for food safety reasons—keep it under 40°F). This will certainly increase the effects of freezing (and would normally be avoided for flavor and texture reasons)
This isn't freezing, but may accomplish the same goal: you could try one of the 40+-blade meat tenderizers. 


Also, as a final note, it turns out that a lo of how we (humans) perceive flavor has nothing to do with the food. The ambiance, how we're feeling that day, etc. affect perceived flavor substantially. Keep this in mind, since you're not using e.g., a salinity meter, its possible you're chasing down differences that aren't due to the food itself.