12:41 p.m. on October 21, 2012 (EDT)
Well, if it were me personally I would carry bear spray.
I have encountered a pack of coyotes on more than one occasion while in the hills. Most recently was last winter during an extended night of solo hiking.
As I was approaching I could hear them yipping away. As I got closer they kept getting louder and louder which meant to me that they were not really worried about my presence being they didn't take off.
I was pretty loud and each footstep made a substantial crunching sound that was easily heard so they definitely were aware of my approach/presence.
At the same time I really wasn't too worried about them either. Coyote/human interaction isn't much of a problem in the ridges here. They are somewhat like blackies, bobcats, etc.
Once they are aware of you they tend to steer clear of people or for the most part keep their distance.
I kept trying to catch a glimpse of them that night with my headlamp but they seemed to just stay out of the range of my beam.
I did see a ton of tracks where there were frolicking around in the snow.
Anywho, it seems as though these coyotes you all are dealing with are not too concerned with human interaction and have found a consistent, viable food source that is easy to obtain.
With that being said I would go on a limb to say that capturing and relocating said animals may not truly alleviate the situation being that now the canines have become somewhat accustomed to this new, wonderful, easily obtained food source.
Relocation may alleviate the situation on the short term but they may find their way back after relocation dependent upon where they are relocated too or this may just transfer the problem to other populated areas.
As much as it saddens me to say this dispatching the problem animals might(notice I said might) be the only viable, long term solution.
I do have a few side arms(.357, .44, S&W .500) and I would say that if you were to encounter an animal or worse a pack under these circumstances you may be up a creek w/o a paddle.
I surely would not use a standard round being chances are very limited in regards to hitting a single, moving target with a single shot let alone multiple targets.
A scatter round(commonly known as bird or snake shot) for a higher caliber pistol might be an option. I surely wouldn't take on an animal the size of a coyote or larger with a .22 loaded with a bird shot round.
This may only injure the animal and an injured animal is a dangerous animal.
My strongest suggestion is to carry a spray with you and let the proper authorities deal with the animals on a larger scale.
If you encounter a coyote(or even a pack) unload the spray on them. Your chances of deterring the animal/animals is much greater and worst case scenario ya may end up with a burning sensation, watery eyes, and a runny nose(dependent upon wind direction, etc.)
Then as soon as possible notify the authorities of your encounter as well as the area in which your encounter occurred so they can get on it asap. It is much easier to locate and pinpoint where these animals might be if notified quickly.
Minutes and seconds can make a huge difference in a scenario such as this in regards to locating or not locating these problem animals and alleviating potential problems later on down the road as I am sure you are aware of.
There are many factors to take into consideration with a firearm.
Once that trigger is pulled the repercussions are not reversible.
Once fired rounds travel and can travel for a great distance. One cannot steer a bullet. They ricochet, hit unintended targets, so on and so forth.
If you are in an inhabited area this risk goes up dramatically.
The effects of bear spray are temporary. A bullet not so much.
Please keep this in mind in regards to your decision on how to approach the problem you are dealing with.
I hope this helps ya out a bit.
My logic comes from growing up hunting, hiking, and trapping(among other various experiences throughout the course of my life.)

I no longer hunt & trap but I still hike and backpack.
...alot