I haven't seen a way to create (or edit) a profile, no.
However, you might be able to do what you want by using a variable for the time value in a profile's time context trigger, like so:
You click the icon with the crossed arrows at the left to switch between a variable and a constant time value for the "from" & "to" settings. The variable should be in "HH:mm" format (24-hour hour, a colon, and then minutes).
Although a similar scheme doesn't exist for a day or date context (e.g. to fire every Wednesday or Thursday depending on the value of a variable), you can get the same result by testing for the desired day/date as the first step in the Task associated with the profile.