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Looking back into Clement Wood's introduction to his Rhyming Dictionary & Poet's Craft Book (1936,) he gives equal sections to Assonance & Consonance. Here's what he says of the latter:

Consonance, also loosely called off rhymes, sour rhymes, and analyzed rhymes, consists in an identity of all consonantal and vowel sounds after the accented vowel: and a difference in the accented vowels. An improvized model would be:

There's a golden willow
Underneath a hill,
By a babbling shallow
Brook and waterfall;

And a mill-wheel turning
Under moon and sun,
As if gently scorning
Time and tide and man.

There can be any combination of end and internal consonance with end or internal rhyme in the same poem, according to the best modern practice : the poet himself being the judge of what form is most pleasing to his inner sense of music, and that of his prospective readers.

He goes on to cite Edna St. Vincent Millay's use of consonantal & assonantal end-rhymes, & Elinor Wylie & Emily Dickinson, & then goes on for four or five paragraphs about how one devises such rhymes. He then says, "There is small adventure in rhyming. . . There is great adventure in the use of consonance, which expands the sound-repetition resouces of the language. It is possible to write a poem where not one of the consonance couplets has been used before. The adventurous will not overlook this."

Some people will tell you that "slant rhyme" is only assonantal. I myself think the term is still somewhat open to interpretation, & that both assonantal & consonantal rhymes are "slant." For obvious reasons. They are both rhymes that bend the definition of rhyme.