Sorry, I don't know the answer in regard to the Jeffery design.

I do know that the Woodward Automatic was a fascinating underlever snap action design that does cock the locks by action of the underlever. It began with a fabulous exposed hammers design:

You can see a two position switch at bottom of action, in one position upon opening the operating lever, the hammers lift to half cock. In the second position the hammers will lift to firing position.

The design evloved into a lovely hammerless snap action; one of the earlier hammerless doubles:

In this configuration the tumblers (internal hammers) are lifted to full cock by the action of the lever. Obviously there were certain reservations about the safety of strolling around with a rifle at full cock, so Woodward incorporated a positive tumbler blocking safety mechanism and stressed safety in its advertising. Here's a copy of an advertisement appearing in 1882:

If you want to see my Woodward Automatic dissected with its guts hanging out, you can look HERE.

In a post above, somebody expressed surprise that underlever snap action rifles would be offered into the 1930's. Well guess what, they are offered today! Butch Searcy offers the underlever as an option with his rifles. I believe it is offered by English and Continental makers as well.

Curl