Fun fact: of the hundred most common surnames in Japanese, all of the ones that use pure on’yomi, and only those, have 藤 as the second kanji. For example, 佐藤, 伊藤, 加藤, 斉藤.
For the rest, you’ll be correct nine times out of ten if you use the kun’yomi, but sometimes there’s half-and-half on/kun names, and sometimes they use nanori instead, which are readings that are used only in names. Also, sometimes there’s invisible particles, as with your 山手 example - 木下 is きのした, 井上 is いのうえ, and so forth.