What was the the value of the Hubble constant at the time of the CMB's 'release' (i.e., 379,000 years after Big Bang)? What about its value roughly 9 billion years after the Big Bang, when dark energy started to 'take over' and accelerate the expansion of the universe?
Is there a timeline or chart somewhere that shows approximate, theoretical values of Hubble's not-so-constant constant throughout the lifetime of the Universe?
This answer to the question “Is the Hubble constant dependent on redshift?” gives the formula (a form of the Friedmann equation) for the Hubble parameter $H(z)$ as a function of redshift $z$:
$$ H(z)^2 = H_0^2 \left[ (1+z)^4 \Omega_r + (1+z)^3 \Omega_M + (1+z)^2 \Omega_k +  \Omega_\Lambda \right] $$
where the $\Omega$ terms are the fractional densities in radiation, matter, curvature, and dark energy, respectively.
Using that, plus the knowledge that the redshift of the CMB is $z \simeq 1100$, you can plug in values for the densities (I used WMAP values quoted here) and get that $H$ at the time of the CMB was about 22,000 times larger than the current value.
That answer also gives a graph of the value of the Hubble parameter as a function of time.