There are four regions to consider here, which are similar in concept but have notably different consequences. I'll first clarify the question about the Hubble sphere and observable universe.
To begin, the observable universe (aka light cone) is the region of the Universe that we can obtain information about. It has a diameter of 93 billion light years. The Hubble sphere, on the other hand, is the region in which objects are receding from us slower than the speed of light. This has a diameter of 28.8 billion light years; clearly, most of the observable universe lies beyond the Hubble sphere. Davis & Lineweaver (2003) provide a good diagram:
Why the disparity? The reason is that photons emitted by objects outside of the Hubble sphere can crawl into regions of space that are receding from us a bit slower than the one from which they were emitted. They can then crawl into regions that expand slower than the last one, and so on. Eventually, the photons will reach regions of space that aren't expanding faster than light, and they can reach us. Hence for the sake of observing, we only care about the observable universe rather than the Hubble sphere.
Now, there is a distance beyond which space expands faster than the light can traverse it. This distance is called the cosmic event horizon. Light from beyond the event horizon will never reach us. Our observable universe grows because more photons come into our view. However, the observable universe will asymptotically approach the event horizon, since light from beyond there can never reach us.
Finally, let's get back to the question about the CMB. The CMB did not originate from the Big Bang itself, but rather an event ~10 seconds after it, called Big Bang nucleosynthesis. Essentially, the Universe was so hot that atomic nuclei constantly fused together. These fusions released radiation that ultimately became the CMB. Even after the Universe cooled enough for nucleosynthesis to stop, it was very dense, so CMB photons would constantly encounter free electrons and scatter off them.
Eventually (378,000 years after the Big Bang), the Universe expanded enough so that CMB photons could travel freely, so after scattering for the last time, they went on a straight path toward our eyeballs billions of years later.
Nowadays, as we look farther away, we see the Universe at older and older times. The point where we'd see CMB photons is called the surface of last scattering, which is slightly smaller than the observable universe. Beyond this, the CMB photons would still be scattering instead of making their way toward us.