Can you pick up from the discard pile and hit in Phase 10?
Yes, you may do that, based on the rules, here's what's relevant:
Hitting:

Hitting is the way to get rid of leftover cards after making a Phase. You make a hit by putting a card directly on a Phase already laid down. The card must properly fit with the cards already down.
Before you can make a hit, your own Phase must already be laid down. You may hit only during your turn. You may hit your own cards, another player’s cards, or both.

Turns:

On your turn, draw one card, either the top card from the draw pile or the top
card from the discard pile, and add it to your hand. End your turn by discarding any one of your cards onto the top of the discard pile.

So you may draw from the discard pile to hit only when:

It is your turn.
You already made a Phase.
The card properly fits the Phase you are adding it to.

Combining these rules means your turn sequence to do this would be:

Pick up from discard pile.
Add it to your hand.
(if you already made a phase) you may hit your own or another player's phase.
Discard to the discard pile to end your turn.

Since you can only hit on your turn, this means you can only use the discarded card from the player before you.