There are a handful of phrases in Japanese that start with 手(て).  Here are a few I learned from a practice JLPT exam (N2):

てまえ on this side (the side closer to me); before
東京から見て、名古屋は大阪より手前にある./Nagoya is closer to Tokyo than Osaka.
私の3歩手前で立ち止まった/He stopped 3 steps before me.
その公園は川の手前にある/The park is on this side of the river.
銀行の手前を右に回る/Turn right just before one comes to the bank.

てごろ moderate; suitable/reasonable
手頃なサイズ/a convenient size
手頃な価格の新薬を開発する/develop affordable new medications
机のそばに手ごろな棚がある/There’s a convenient shelf next to the desk
手ごろな値段で洋服を売る/Sell clothing at reasonable prices

ていれ maintenance; repairs; caring for; tending
この家は手入れしないことには住めない/This house can’t be lived in unless it is repaired.
庭木の手入れが済んだところだ/The trees in the garden have just been trimmed

てあて allowance/compensation; treatment/medical care
応急手当(おうきゅうてあて)/first-aid treatment
手厚い手当てを受けている/receiving excellent medical treatment
看護人の一か月の手当て/a nurse’s monthly pay
出張手当て/travel allowance

てあつい courteous; cordial; warm
我々は手厚くもてなされた/We were hospitably received
遺体は手厚く葬られた/The dead body was buried with all due respect.
手厚い看護を受けた/I was nursed with great care.

Examples from Yahoo!辞書, 英辞郎 on the web, N2 1994 Exam

If you can think of some more hand-related words, please comment. Cheers!