What literary device is it if a word begins with the sound the previous word ends in? such as 'doubt' and 'truth' in: Doubt truth to be a liar.
I think you are looking for consonance. It is the consonant version of assonance, and alliteration is a more specific type of consonance.
However, I do not think I would characterize the end of "doubt" as having the same sound as the beginning of "truth".