You seem to have a development problem. I know this because you said:
Making a fix sounds easy, but often times there are other changes that have been made to the code base. And the fix needs testing before it can realistically go out.
You need to isolate you production releases in your version control system (you are using one, right?). Create a branch for your release. Bugs that are discovered in the production release get fixed in that branch (and the fix merged back to the main development line). Bugs that are fixed in main development have the fix brought into the branch if practical (or a parallel fix created if not).
This pretty much guarantees that fixes to the running production code will be tiny fixes, and easy to test (therefore quick to roll out if there's a problem), and hopefully makes them less disruptive.
Outside of that here's the best advice I can offer you:
Have some kind of internal test cycle.
Don't release code that doesn't work, or at least inflict it on a small set of users first.
(StackExchange uses meta.stackoverflow.com as their scratch monkey.)
Cut releases when you're reasonably confident they work
(The above-mentioned branching)
Fix bugs before adding features
(Liberally stolen from The Joel Test)
Don't be afraid not to release something if it's not ready.
Better not to ship a patch on "patch day" than to break the universe.
Don't be afraid to roll back something if it's not working.
(Corollary: Always have a back-out plan!)
Set a release schedule you can keep.
Once a week is HARD. Once a month is tough if you're doing more than bug-fixing. Once a quarter is pretty easy.
For reference, my company releases software the weekend following the 15th of every mid-quarter month (i.e. the weekend after February 15, May 15, August 15, November 15). For various regulatory reasons our testing cycle is pretty rigorous, and this is as fast a pace as we can maintain while keeping software quality where we need it to be.
Emergency bug-fix patches are released as-needed, usually on Fridays (to give us a weekend where we have relatively low volume in case the fix breaks something else).