The Ring of the Serpent is a Boss Relic which can only be obtained by the Silent. It replaces the Ring of the Snake and draws 1 additional card at the start of your turn instead of 2 additional cards but only at the start of combat.
Since it replaces Ring of the Snake, you actually lose 1 card draw on turn 1 ( 7 vs 6 cards ), card neutral on turn 2 (12 vs 12) and only starts to be positive on turn 3 (17 vs 18).
If a deck can confidently deal with the loss of early card draw, (Bag of Preparation, Acrobatics) it may be worth considering to take it so you can better handle longer fights such as bosses. However, other Boss Relics are likely to be of more value while also keeping your Ring of the Snake.
Weekly Patch 56: The Collector
Gambling Chip relic now triggers after card-draw based relics such as Ring of Serpent.