When “Serpentine Princess” is returned to the Deck by an effect occurring in a chain, the effect of “Serpentine Princess” starts a new Chain after the current Chain resolves completely.
If the effect of “Fiber Jar” places “Serpentine Princess” from the field into your Deck, the effect of “Serpentine Princess” is activated and will be applied after the effect of “Fiber Jar” has finished resolving.