If the experiment will be on an egg you bred yourself, and the egg has never left your scroll, you can check your Action Log for the time (EST) the egg was bred. Its ToD will be exactly 7 days later.
Spend less time refreshing
Instead of spending a whole hour refreshing to catch the dragon's hour-change, start by narrowing the options. Spend the first hour refreshing once every 10 minutes, until the dragon's time changes. Record the refresh time prior to the change, and the refresh time after the change (e.g. 9:20 and 9:30). Then in the next hour, find the exact moment by refreshing only during the 10-minute interval in which the change ocurred.
Avoid time zone confusion
When you want to share the ToD with other experimenters, use "xx" in place of the hour. If it is currently 3:15, and you have an experiment that dies, according to your clock, at 6:35, tell others that it dies "at xx:35 in 3 hours" instead. That way, everyone can easily adjust the hour to their own time zone, rather than wondering if 6:35 for them is at the same time as 6:35 for you.
Always unfog on time
Once you have found the ToD, subtract 5-10 minutes and record the result as your unfog time. If you have a bad memory or would like a reminder when it's time to unfog, set up an alarm to get your attention.