Chicken are not my expertise but raising puppies is. Not to worry Cocci does not cross from human to dog or to chicken. Each species has it's own form of cocci. Cocci always shows it's 1st signs in 1 or 2 puppies in a litter.....lethargy, runny stools(often with blood), and quit eating and even vomiting. It comes on over-night and dehydration is what will kill them faster than anything else. You did good with the water drops. That is what probably saved your chick. With puppies they recommend sugar water or Gatorade to keep the electrolytes up, but I'm not sure about chickens. Always treat the whole litter, or in your case, the whole flock for cocci. If one has it, they all have it. It is just that little chick you noticed might have been a bit more stressed than the others and came down with it first. It spreads quite quickly and the young are the ones who will get sick from cocci because their immune systems are not working 100% yet. It is a must to always add probiotics or plain white yogurt(good bacteria) when administering meds of any sort. Cocci meds will kill all bacteria (good and bad) so you need to introduce the probiotics to put good bacteria back in the digestive tract. I always give cocci meds (Albon for puppies) and probiotics at the same time, then continue the probiotics a few days after the 7 day cocci treatment. Cocci is a tough bacteria to kill. It may come back. You can follow through with the 7 day treatment to all of your chicks again just as you said or take a fresh fecal sample to the vets 5 days after the 7 day treatment and have them test it to see if cocci is still present, then you can decide if I want to do another round of cocci treatment. Eventually they'll get big enough and their immune systems will fight it off on their own. Stress is the #1 factor followed by shots, chilled, or rehousing to bring an onset of cocci. It is known as an opportunistic bacteria. In other words, it is almost everywhere but just waiting for the host to get stressed and then it grows ramped with the host's immune system unable to keep it in check.
Don't bother trying to kill cocci with bleach or any type of cleaner. Nothing kills it but fire or hot dry full sunshine 80+ degrees for weeks on end and no rain. It will live for years in the soil and many hosts will be carriers for life. Damp cool puddles of water does seem to spread it way faster. You sound like you have everything you need including the antibiotics for secondary infection, which I would give as a precaution if need be. Why this is needed for puppies, and I'm sure for chicks too, is the intestines are so flared up(that is the blood your seeing in the stools) that the intestines are literally bleeding so this means open blisters are in the digestive tract. The same place their poop is. This is the leading cause of septicemia in animals. The bacteria then gets in the blood stream via the open wounds in the digestive tract = blood poisoning(septicemia). I guess it depends on how bad the bleeding is in the poop or if recovery takes longer than a few days for me to decide to add antibiotics and what kind. Again. the probiotics are a must! Focus on getting that digestive tract working smoothly and unirritated as quickly as possible so they don't have a chance of getting septicemia.
With quick action, which you have done, and proper meds you shouldn't loose a chick. I've never lost a puppy to cocci or secondary infection. I've had a lot of scares but no losses.