I don’t know how to put it better than that. For example, …unk:
bunk
chunk
dunk
funk, flunk
gunk
hunk
junk
klunk
lunk
monk
punk
sunk, stunk
The …unk rhyme can be started with 12 of the 26 letters of the alphabet (if you accept klunk:)). Is there any rhyme that can start with all 26?
At, bat, cat, drat, fat, gnat, hat, mat, pat, rat, sat, that, vat
there’s half the alphabet, anyway
Well it all depends on how far you want to go to rhyme and what you’d consider a rhyming word.
For example, would you think that “debunk” would go into your example?
But that aside, I think that either -ight or -ed would probably take the cake for most rhymed words.
-ow (read: owe) would work out well and probably be the most rhymed if we were, again-- like I asked above, including very LONG (yet still rhymable) words like “although” and “arrow” and any other very, very long word as long as it ended in that sound.
If that doesn’t count though, I’d have to stick with -ight or -ed (these are just guesses, however).
Well, I don’t know, and it’s sort of inherently fuzzy, but here’s a large one (I assume you only want one syllable words):
Aye, bye, cry, die, eye, fly, guy, hi, I, lie, my, nigh, pie, rye, sigh, tie, vie, why
That’s 18, and maybe people can fill in some for some of the letters I missed.
I vaugely remember an alphabetical rhyme book. I think it was pink, about 100 pages, and hardcover. It purportedly was the “biggest rhyme book ever” (or something to that affect). Since it’s alphabetical, it should have the answer.
This makes me wonder again if the OP would include any rhymed words as long as they ended in the same rhyme. Because then the list would just go on with words like: Verify and multiply and stupify and beautify and etc.
Oh no, you’ve caught my quote in the flux between edits! That means you’ve captured my soul, or something. 
As you can see, in the latest version, I note that I’m working with a one syllable restriction.
(Also, none of the new words you propose would actually add new letters to the list, but, yes, on the same principles, we could easily blow it up)
Well okay. In that case I see your one syllable rule and raise you a spy, spry, and ply!
Yeah, and dry, fry, the dreaded “gry”, etc. We could characterize by initial consonant clusters, and I think this class would still do pretty well. But I think the OP is concerned only about characterization by initial single letters.
Okay, I gotcha. I get what you’re saying and it does seem that way rereading the OP.
bay, bray, cay, clay, day, dray, fey, flay, fray, gay, gray, hay, jay, lay, lei, may, nay, neigh, pay, play, prey, pray, say, stay, stray, sleigh, slay, tay, tray, way, weigh.
That’s 14 distinct initial letters. We can toss in “a” and “eh” and “ray” and “yea” to make it 18. And there’s the word “quay” for 19, but it’s sort of uncommon. And perhaps you’ll count “kay” (as in the letter), to make it 20. Just missing I, O, U, V, X, and Z, then.
X-ray? 
(I hate how that’s the example of a word that starts with “X” in so many alphabet books. It’s even the X word in “radio code.”)
Islay.
Okay
Quay is pronounced like key, so that doesn’t count.
Would ‘Oy Vey’ count for V?
Continental Europeans often pronounce ‘They’ as ‘Zey’.
There’s also “Y”. And although they aren’t new initial letters, it’s worth considering different initial phonemes, like chai, shy, thy, thigh.
But those don’t rhyme with the one-syllable “eye” words… At least they’re not what are known as “perfect” rhymes, which is what I assume most people talk about when they talk about rhyming in general. A typical percect rhyme is one in which the last stressed syllable and any following unstressed syllables, contain the same sounds. For example, “Audible” and “laudible” are perfect rhymes, while “audible” and “kibble” are not.
As I noted, I’m working under a one-syllable restriction.
That’s one pronunciation, but according to dictionary.com it can also be pronounced as “kay”.
I considered that, but decided it wasn’t really reasonable to classify “vey” as an English word.
Yes, but they certainly don’t spell it that way, and it seems to me this thread is all about spelling, having its focus on initial single letters rather than initial phonemes.
What about plain old “ee”? There’s bee, brie, dee, eee, gee, glee, he, key, lee, me, knee, pee, re:, see, she, tee, thee, tree, vee, we, ye, and zee.
air
bare, bear
care, chair
dare
fair, fare, flair, flare
glare
hair, hare,heir
lair
mare
pair, pare, pear
rare
scare, share, snare, spare, square, stare, stair, swear
tear, there, their, they’re
wear, where
George Carlin fan?