The Big Bang model is the current large-scale accepted and tested theory. However, several new physics models or theories propose that there could be a previous Universe. To what extent can be test this hypothesis? I mean: are the CMB polarizations and other fingerprints like the likely cosmic neutrino background or the gravitational waves from the beginning of the Universe the only tests of pre-Big Bang theories? P.S.: I know this question is highly speculative.
Cmb and all other methods you mentioned above are not for discovering pre big bang physics at all. They are for understanding the event of the big bang, so... post big bang not pre. Unfortunately there are no known ways (not even theoretical) to test whether or not another universe preceded our own. Besides I seriously doubt there would be any detectable remnants of that universe and how would we distinguish what was from ours and what was from the other hypothetical universe.
One pre-Big Bang theory that is potentially falsifiable is Nikodem J. Poplawski's "Cosmology with torsion", described in numerous preprints (written between 2009 and 2020) that are available free on Cornell University's Arxiv website, as well as in such reputable journals as Physics Letters A.
His cosmology provides for the creation of local universes, each causally-separated from any other, that would include our own "Universe", as well as its "observable region" (the part of it which we can actually see, at least as that part was--like everything we see--in its past).
It is based on the purely relativistic Einstein-Cartan Theory, developed through conversations between Einstein and the mathematician Elie Cartan in the late 1920's. ECT is (reportedly) more complex, mathematically, than Einstein's 1915 theory (General Relativity), although Poplawski's cosmology is as mechanistically simple as any I've ever heard of.
ECT requires that fermions have a (tiny) spatial extent, and the cosmology is based on the interaction between virtual fermion/anti-fermion pairs and the vastly-larger fermions of any rotating star that's collapsing gravitationally (after the depletion of its nuclear fuel has left it without the radiation pressure needed to uphold its own weight). The "Event Horizon" of the star's collapse, propagating outward from its center, permanently separates the virtual particles of many fermion/anti-fermion pairs from each other, leaving one partner of each pair free to remain only virtual while escaping toward infinity, while the other one is materialized into reality by the intensity of the star's collapsing gravity, in a sort of reversal of the subatomic processes unfortunately familiar since Hiroshima.
When the newly-materialized fermions come into contact with the vastly larger ones of the star itself, the spin of those stellar fermions propels many of them outward into the outer reaches of the collapsing star's original volume, where, as their momentum decreases, they form a new "local universe", whose shape Poplawski analogizes to a three-dimensional version of the surface of a basketball. Per Maupertuis' "Principle of Least Action" (from which General Relativity can reportedly be derived), this development can be considered as spatial expansion, rather than relative motion.
The ball-shaped region we see through our telescopes is only a small portion of Poplawski's LU, but, in a 2015 collaboration with Desai titled "Non-parametric reconstruction of an inflaton potential", he accurately describes its correspondence with our Cosmic Microwave Background radiation, as well as identifying his theory as a version of cosmic inflation (quasi-exponential spatial expansion), without any need for the specialized "inflaton" particles whose identification has become very problematic for inflationary theories relying on a hypothesized scalar field.
His theory would also appear to have some potential for explaining the huge disparity (at 121 orders of magnitude) between the observed and theoretically-hypothesized amounts of vacuum energy, as it would provide for a temporal multiverse endlessly replicating itself (albeit generally only approximately) on sequentially-decreasing spatio-temporal scales.
It qualifies as a scientific model, rather than a purely philosophical or spiritual one, because its hypothesized causes are falsifiable: They depend, for each of the local universes of his temporal multiverse, on the rotation of its parenting star. Most stars rotate appreciably, and all stars have generally been expected to have at least some faint residual rotation, but Poplawski's theory might reasonably be considered less than probable if the observable stars did not, overall, have a prevalent direction of rotation. In fact, such a prevalence would probably be minimal, as studies on the subject practically alternate between favoring such a prevalence and discrediting it: The most recent I have found, at
https://hal.archives-ouvertes.fr/hal-02045149v3/document, dates from January of 2020, and favors that prevalence.