How big is your greenhouse, how many plants do you have, and what is your water quality like before your well water enters a water softener?
Regarding water quality, there are many things that could be measured, but the most important is total dissolved solids (TDS). The cheap little TDS meters are less accurate than most people would think; you are better off submitting a water sample to your local "county agent", or you might be able to talk someone in the local water department into measuring "Electrical Conductivity" (EC), also known as specific conductance, or just conductivity, for no cost or very low cost. TDS can be estimated from an accurate EC number; make sure the lab reports the units of measurement (often micro Siemens per centimeter (ÁS/cm) or milli Siemens per centimeter (mS/cm). I can do the conversion easily if you get the lab number.
If your pre-water softener water quality is OK, you may be able to draw water from your well before it enters the softener (should be an easy plumbing job).
If the quality is not OK, a watering system may be possible to rig from your rain barrel. Possibly a fountain pump or light duty submersible. Some options are available at Pumps – Fountain Pumps & Water Pumps