Typically, I divide these kinds of requests into 2 camps:
Fix the bug, not the whole code. In this case, I'd tend to read the code and see if I can easily put in the fix without making major significant changes, e.g. adding 5 lines of code or less. This is where the changes made are like a surgical strike that does minimal impact to achieve the desired result.
Tear down the code and put up much better stuff. In this case, there isn't a good little way to do the change or there are lots of little changes that want to get bundled up into a mini-project where digesting all the code is needed and this isn't a quick minor change. Taking an ASP site and putting it into the latest .Net framework would be an example here.
Determining which kind of change something is requires a bit of practice and experience understanding what someone else may call, "One more thing if you could..." that may be a huge deal but they don't see it that way,e.g. could we add a little module that records any error and sends me an e-mail when something bad happens that may not be so easy after all the other requirements get worked out like what if the code can't send a mail message what should it do.
