The reason why the Hubble constant measurements (measurements of distant redshift) keep turning out different by different methods is that there is no Hubble constant. It isn't a constant. The universe is not expanding at a constant rate in every direction at every distance at every time. And even if it were, it wouldn't look like a constant.
If the big bang is true, then the expansion cannot be constant. If the universe is a solid sphere and not a beach ball with a thin shell, then the expansion cannot be constant. If general relativity is true, then the expansion cannot be constant.
Big Bang: If the universe originated in a big bang, then the expansion cannot be the same everywhere. The random and contrary expansions would have canceled each other, as almost everything ran into almost everything else.
Spherical universe: If the universe is a solid sphere, then the expansion cannot be equal, because part the outer edges would have been expanding inwards. This is exactly the same problem as the big bang. The only way for uniform expansion to work would result in a hollow universe with a thin shell of galaxies. And the expansion would only look uniform side to side. There would be very little (if any) expansion from front to back.
General relativity: The farther away things are, the faster they appear to be moving away. This effect is subject to the Lorentz factor comes into play. This is what we actually see with the redshift data from distant supernovae. Their redshifted light closely approximates the sine wave of relativity. And the farther away they are, the more closely they approach the redshift of the CMB - the most ancient and furthest light we can see. Even if expansion were constant (which nothing indicates), then it wouldn't appear constant.
Summary: Space is not expanding. The stuff in space is expanding from a central point, the origin point of the big bang. There is an outer edge, which is moving away from the center at very nearly the speed of light. The stuff near the center is barely moving at all. But no matter where you pick, the universe always looks about the same, due to relativistic effects.