When the pressure is on: slow down!
This may seem counter-productive, but it isn't. Mistakes will always be made. Mistakes are more likely when you are in haste.
When the pressure is on that is often because (some) client(s) can't continue their work/use of your software until a problem is fixed, or a problem needs to be fixed before the next version can be released.
All the more reason to take it slow(er) when the pressure is on. If only because the stakes are even higher than usual as the lashbacks (clients losing trust in you/your software) are more severe when you get it wrong.
So, whatever you do, when the pressure is on do not skip important steps.
Even if there is a manager panting in you neck. Just tell him to go away and let you do your job or it is going to take even longer. If only because (s)he is distracting you...
When we are faced with a situation that requires a hot fix because there is a problem that is preventing an entire company from using our software (which they have come to depend upon), we no longer "ship" the hot-fixed version until it has been tested by QA. We feel it is better to have them wait another couple of hours or even a day, than to provide a "solution" that needs to be fixed again... (and again, and again...)
I can't remember his name, but I think it was a Japanes person/philosopher who said that the busier you are the more often you need to lie down for 5 minutes and consider what you are and will be doing next. I have found it helps me keep my priorities straight and prevents me from chasing after someting that doesn't contribute to my goals. I have found it also helps in my software development to keep me focused on the end-goals (user must experience value from our software) and not get caught up in a "let's put out this fire as quickly as possible" rush.