Hi, I can't speak to the packing on that spigot, but I wanted to mention that you're using those quick disconnects backwards. The female end, the part that has the sliding collar, should be put on the spigot and on the far ends of any hoses. The male end, the grey part shown in the photo, should be put on the near end of hoses and any attachments, like sprayers, sprinklers, fertilizer sprayers, nozzles, etc. The female end, the part with the sliding collar, has an automatic shutoff valve inside of it. The idea with using the quick disconnects is that you can uncouple an attachment or a hose segment, without having to go back and turn off the main valve at the spigot. In fact, if you have good washers in everything, and good quality hoses and QD's, you could turn on the spigot at the wall and just leave it on all summer. Water hose QD fittings should be used in the same way that air hose QD fittings are used, if you're familiar with those.
BTW, I know *that* particular male QD fitting won't go on the near end of a hose, but that's a reverse QD fitting for oddball situations. The common setup is that the female QD fitting also has the female hose threads, and male QD fitting has the male hose threads.