The small thumbnail attached to this post is the Zenith radio I mentioned in my last post that looks like a miniature console, and is probably the predecessor to yours. Grabbed the picture off eBay a few minutes ago. As I said, I guess there are still a few of these sets still in existence--this one, and yours of course, are living proof of that. I'm happy to see that these sets are being saved from certain death in landfills, as most of them have only minor problems the owners don't want to or cannot fix themselves.
Hmmm.
If the speaker in your set rattles even at very low volume, I'm not sure what could be causing the noise. I'd check between the speaker and the front of the cabinet first, in case there may be something stuck in there that moves when the speaker cone does (as I mentioned in my previous post). That failing, I'd look at the cone itself, as it may be rubbing against the speaker frame at some point; since the rattle is present at all volume levels above a whisper (!), this seems very likely.
I honestly do not think the rattle is being caused by anything else; in fact, it would surprise the dickens out of me if a chassis fault was causing this problem. I can't imagine how such a fault could cause a speaker rattle, unless the output transformer, if there is one, is intermittently breaking down to the frame and arcing; you would hear a noise that sounds like static, even on strong stations, if this were the case. However, if the radio basically sounds good as you mention, I would check the cone for starters. I wouldn't be surprised if that's the trouble.
I would also check the cone for tears, rips or other damage. A torn speaker cone can cause very exasperating rattles whenever it moves; also, if the speaker has been repaired by the former owner, the tape used to patch the rip could well have dried out and become detached, again causing a disturbing rattle at any setting of the volume control.
Good luck. That radio is well worth saving, as it was probably, even likely, one of the last quality radios made by Zenith before they went out of the radio business for good. The very last radio Zenith made that could be classified as "high fidelity" was built in 1982, and was the model (I forget the model number) that had a tuning meter and separate bass/treble tone controls. This was also the end of Zenith's good quality stereo systems, IIRC; I had a 4-mode integrated system, also made in '82 or maybe a bit earlier, that had tuned-port speakers and 5 watts per channel, but it was made in Korea, probably very shortly after Zenith's radio and audio division split the Windy City for the Pacific Rim.
This small set (Zenith's last table radio) also had what Zenith referred to as a "bass booster" on the back cover, although I think this was little more than an air chamber that contributed little (if anything) to the overall sound quality. This radio looked much too small, IMHO, to be a true high-fidelity set; the speaker was probably too small for good bass response as well.