I have a mechanical dial and I would like to make the first input on my combination as soon as I lock it up so that I can get in quicker next time.

It might make it a little easier for someone to "pick the lock" but what are the odds of a thief actually knowing how to do that?

If someone is going to break into my safe it is most likely that they will just use an angle grinder or pry bars right?

You'll have no way to know if it's been tampered with (someone spinning the dial) until you try to open it and it doesn't open.  If you live alone and are 100% sure no one will mess with it, then sure, knock yourself out. Otherwise, probably just easier to resign yourself to entering the full combo.

For that reason, I have several quick-access safes around the house.

ETA: Are you just being lazy, or are you looking for a few-second tactical advantage in a potential bad situation?

A lot of people (and businesses) park the dial on a number whether it's their first number or not.  When opening the safe the next time the first thing they do is look to see if it is still on the number where they left it.  If not, it's an indication of tampering.

There would be a security risk against a skilled manipulator.  I don't see any risk against 99.99% of people who may be trying to break into the safe.  Most of those types aren't attempting to dial it open anyway.

Just for that reason. Or if the house caught on fire and I needed to rescue the important household members.

Only me, my wife and daughter and neither of them have any desire to open my safe.

It may be just me but I don't put rapid deployment items in the safe. I use an alternate method to secure those items.
Safe is for security not speed.