Phase 10 Going Out - Do you keep picking up?
Before making your phase, your turn structure is like this:

You MUST draw a card from the deck or discard pile.
You MAY make your phase and put it on the table.
You MUST discard a card onto the discard pile.

When making your phase you MAY immediately add any number of your own cards to other phases on the table. Afterwards, you MUST discard a card to end your turn.
Then after making your phase, your turn structure is like this:

You MUST draw a card from the deck or discard pile.
You MAY add any number of cards from your hand to the phases on the table.
You MUST discard a card onto the discard pile.

The round ends immediately when a player gets rid of their last card.