Gin Up

From BoardGameGeek:

Gin Up is a climbing-shedding game for two players inspired by the classic game of Gin Rummy. Players will take turns drawing and discarding until one of the players decides they have a strong enough hand to knock. Once a player has knocked, the game pivots to a climbing-shedding phase, where both players are trying to be the first to shed out. The game can be played with a standard 52 card deck. Hands of 13 cards are dealt to each player, and one card is turned up from the deck to start a discard. Players alternate drawing a card to their hand, either from the discard or the draw deck, and then choosing to either discard a card or knock. Once a player knocks, there is no more card exchange, and the player who knocked will initiate a climbing-shedding phase. During the climbing-shedding phase, players will play successively higher combinations of cards (similar to melds in Rummy games), trying to empty their hand of cards before the other player. The player who knocked gets the advantage of going first, but the player who didn't knock gets an advantage too: they may, instead of beating a previous meld, add on to it if they have the appropriate card. The player who manages to shed all their cards first scores points based on the cards remaining in their opponent's hand.