The simplest way to achieve a double seal is with induction cap sealing.
All that is needed is a foil liner to be placed inside your existing cap before it is screwed onto the filled container. (It’s common to buy caps with the liners fitted direct from your cap supplier, you don’t have to fit them yourself).
The capped container passes under an induction field and the instantaneous heat melts the plastic on the seal creating an air-tight seal.
The hermetic seal is so strong, it is more likely the container would burst rather than the seal fail.
For a more detailed look at how a heat sealing machine can help you prevent e-commerce packaging leaks, take a look at this video from Selig, a global leader in supplying induction sealing materials for consumer packaging.
Amazon show an induction sealed container as an example of a non-breakable, double-sealed container.
An induction seal will easily pass Amazon’s 125cm drop test.