I would take her in earlier if possible. Sounds like she has an upper respiratory infection and probably having a difficult time breathing and swallowing the stuff dripping down her throat, which is what you're describing as hiccup-like symptoms. Some other signs to look for are runny eyes and a runny nose, as well as lack of energy and lack of eating.
We just went through this with our new kitty that we adopted just after Thanksgiving. In the meantime, to help her out, keep her in a warm environment, but not too stuffy. Confine her to one room if at all possible, with her food, water, and litter box. Invest in a vaporizor and run it where she sleeps. This will help break up the gunk in her head/nose. If you can't get or don't have a vaporizor, run a really hot shower in your bathroom and take her in there several times a day to sit in the steam. (Don't put her in the shower or you'll have a different problem!) Make sure she's eating very well and drinking LOTS of water. If her little nose gets stopped up, she'll quit eating because she can't smell her food. That's not a good thing at all!
Just so you'll know, it's very common for kitties at shelters to have upper respiratory infections, so it's not anything you've done or not done to cause this. Also, the stress of the new environment and just the change of being in a couple of different places in a short period of time can aggravate the situation and cause a flare up of an URI.