When you discovered the problem you should have capped the hose bib when not in use. Every hardware store has metal caps for exactly that purpose. You might try an electrician's "fish tape" to clear the hose bib but BE CAREFUL. You could get one of those big suction gizmo's and fill it with hot water to see if you can loosen the stoppage and then let the system water pressure blow it out.

Ginny try just to snug up the packing nut behind the wheel first.

My preference would be neutralizer (2 in parallel, or one larger)==>iron removal (there are a variety of options - often the best choice is the one your local treatment folks are most familiar with, but I'll post a list of options to consider)==> GAC (you may find it unnecessary with proper iron filtration) ==> softener. Some iron removal options: Softener: You are already living with this method - A softener with specialty resin is capable of removing dissolved iron, up to 7 ppm (mg/L) but asking a softener to remove iron at those upper limits is really pushing it. In addition, a softener will become fouled with particulate iron over time. I don't recommend this method but, in your case only, it has the benefit of already being installed. Oxidation/Filtration: An oxidizer such as ozone, air, or chlorine may be used to react with the iron and force it to become particulate iron that can then be removed via filtration. This will also remove most sulfur compounds. A typical setup would involve the oxidizing unit, a holding tank, then a media filter. This type of system works quite well, but takes some expertise in sizing - you would want a water treatment pro that you trust to help you with this option. Oxidizing filtration media: This type of treatment consists of a sealed tank filled with one of several media. Water passes through and is oxidized and filtered by the media. Most will remove sulfur compounds as well as iron. These are relatively easy to operate and what I would recommend for the average homeowner. There are several different media that can be used: Manganese greensand: water runs through for treatment. The media must be regenerated with potassium permanganate. Care must be taken with dealing with potassium permanganate as it readily dies organic material, such as your skin, a purple-brown color. Some people are quite comfortable dealing with the chemical; others are not. Synthetic greensand: This is essentially the same as option (a) but consists of a coating of greensand on a silica sand core so does not require as much backwash flow. Service flow rate is 2 - 5 gpm/sqft. Backwash flow rate is 12 gpm/sqft. Birm: This media acts as a catalyst to force oxidation of iron. While it does not need to be regenerated, it does need fairly high dissolved oxygen in the water. If your water does not have adequate dissolved oxygen (and it probably doesn't since it is well water), air injection would be necessary prior to the birm. Additionally, birm requires a minimum pH of 6.8. Service flow rate is 3.5 - 5.0 gpm per sqft. Backwash flow rate is 11 - 20 gpm, depending upon water temperatures and desired bed expansion. It is old tech so lots of water treatment pros know how to use it. Pyrolox: an ore that oxidizes then filters the iron out. It does not need regeneration, but needs to be backwashed (to rinse out the iron) at a high rate. pH range is 6.5 - 9.0. This type of filter works very well, but backwash is critical. Service flow rate is 5 gpm/sqft. Backwash is 25-30 gpm/sqft. Backwash daily. Terminox: Similar to Pyrolox, but a proprietary formula . It does not require as much backwash flow rate and is more resistance to a low pH. The particulars are only available from the company that sells it. Backwash daily. Filox: Also similar to Pyrolox. pH range 5.0 - 9.0. Backwash flow 12-15 gpm/sqft. Service flow 6 gpm/sqft. It must be backwashed daily. This one may be a good fit for you as it can take the lower pH. If you choose this option, your treatment order could be: Filox==>Neutralizer==>GAC (maybe)==>softener

The tan knob at the top looks like the vacuum break. Here's what I would do at this point... I'd turn the water off at that shutoff, and then replace the entire assembly (frost proof sillcock). You can get the whole thing for $30 at big box hardware. Note that if you go this direction you need to unscrew the sillcock from the copper pipe while holding the copper pipe from turning (so two wrenches). Frost proof sillcocks come in various lengths - match what you have. Also match the end that connects to the plumbing (is probably 1/2" male threaded). While you are there at the hardware store you will need a small roll of teflon tape to wrap on the pipe threads before you assemble again to prevent leaks. You will also need a can of "Great Stuff" or similar foam. With the new sillcock connected and properly in place, foam the opening around the sillcock from the inside so it's sealed up and keeps out air, bugs and critters. You don't want foam outside, so the new sillcock should be positioned tight against the house. If foam does get outside, don't mess with it until it's dry... then cut it / clean it up.