So - you've just got the one heat exchanger? That is being used to both charge storage via wood, and draw from it via the oil side? And when charging, the flow goes wood-coil-oil-return to wood?
YES
I don't see an issue with adding another circulator, say between the coil and oil-supply.
I would put it between supply and return lines, between wood and storage, on the input side.
Or do you mean a pump on the output side of the coil?
However, I'm not sure what you have for boiler protection at the bottom of your bypass T. That might close up before your storage gets to your desired low & dead-head your circ.
I just keep in warm. And the rise in wood heat is so slow it's never been a problem.
And is there a means in your controls to shut off the circ if storage temp drops below that desired low?
The storage is completely passive. Just a means to recover some BTU that would have been lost without storage. The house is already running a smoother temp with shorter calls for heat.
You might need to add on another strap-on stat, or something like a Johnson A-419.