Yes, on reaches, a properly trimmed staysail will increase the "pull" of the genoa/yankee by creating a bit of a slot.
From Sail Magazine:
A Cutter that Cuts It
For many cruisers, a cutter rig is the one that works best—so long as the staysail is cut for windward work, fairly flat with its draft well forward. A staysail also needs a good sheet lead. Sheet tracks and leads for many staysails seem to be placed more for convenience than effectiveness and often fail to take into account the staysail’s dual role.
On most cutters, the staysail is used with a larger headsail primarily when close reaching. When sheeted inside another headsail, the staysail must be trimmed more tightly than its companion, requiring a track fairly well inboard. However, when used by itself as a heavy-weather windward sail, a staysail may require a more open sheet lead, especially as few cruising boats are capable of pointing really high. At these times it is better for the track to be farther outboard.
The problem with staysails is that most are so small they are close to storm-jib size. If the inner forestay is attached to the mast at the upper spreaders, the sail may also have a short luff, and it is the luff of the sail that powers a boat to weather. For this reason, many cruisers have adopted what some call a Solent stay, which is attached to the mast and deck only a couple of feet behind the headstay. This allows for a staysail of sufficient size to power the boat upwind and in most cases does not require the running backstays used with most other cutter rigs. The downside is that you do have to change sails if you actually need a storm jib. T.J."