Is there a word that means "almost rhymes"?

Solution 1:

It is called half-rhyme.

...is a type of rhyme formed by words with similar but not identical sounds. In most instances, either the vowel segments are different while the consonants are identical, or vice versa. [Wikipedia]

It also has a bunch of other names:

• near-rhyme
• lazy rhyme
• approximate rhyme
• inexact rhyme
• imperfect rhyme (in contrast to perfect rhyme)
• off rhyme
• analyzed rhyme
• suspended rhyme.

Solution 2:

I think assonance is close to what you are looking for: (from TFD)

• similarity of sounds in words or syllables.

• resemblance of sound, especially of the vowel sounds in words, as in: "that dolphin-torn, that gong-tormented sea" (William Butler Yeats).