Assonance is used sparingly in "Climbing My Grandfather." This spare but precise usage helps the poem evoke its delicately calibrated imagery.
A good example is in line 7. Here, the poem employs the long /a/ (italics) and short /a/ (bold):
to an earth-stained hand. The nails
The assonance is cleverly used here. The line describes markings on the grandfather's hand, evidence of some aspect of how he lived his life (perhaps gardening). The line is literally stained by these similar sounds, conveying the image in sound.
Another example is the shared /i/ sound is line 9:
the skin of his finger is smooth and thick
The sound here is short and quick, evoking a kind of narrowness that fits with the description of a finger (and with "smooth" skin).
That example contrasts well with the longer /e/ vowels in line 18:
Refreshed, I cross the screed cheek,
The speaker here has to travel horizontally rather than vertically, and the long vowels give a stretched sound that, read left to right, seems to match with this horizontality.
Another example of assonance is in "birds circle" (line 25). This is a pretty subtle moment, but perhaps helps to evoke the circular movement of the birds by repeating the same sound in quick succession.
In general, the poem uses assonance to capture the carefulness with which the speaker climbs the grandfather, observing every image with precision and traversing as gently as possible.