Did the Gautama Buddha say something about this?
Yes he did! It is in the Satipatthana Sutta.
Samahito sampajano, sato Buddhassa savako;
vedana ca pajanati, vedanananca sambhavam.
Yattha ceta nirujjhanti, magganca khayagaminam;
vedananam khaya bhikkhu, nicchato parinibbuto'ti.
meaning:A follower of the Buddha, with concentration, awareness
and constant thorough understanding of impermanence,
knows with wisdom the sensations, their arising, their
cessation and the path leading to their end.
Notice he says "constant thorough understanding of impermanence".
This is the equivalent of your "be mindful at all times" or
"have your mind on your meditation topic at all times"
Lastly I want to say this is the most important thing anyone can communicate to you about meditation. Never lost your moment...
Know this truth to the exclusion of every other piece of dhamma knowledge you know. Forget suttas, forget teachers, forget jhana, know only "constant awareness".
If you do this you will travel at a very fast speed.
Here is Buddha talking concretely about how fast one may travel:
Verily, monks, whosoever practices these four foundations of mindfulness in this manner for seven years, then one of these two fruits may be expected by him: highest knowledge (arahantship) here and now, or if some remainder of clinging is yet present, the state of non-returning.[28]
(I should add that this quote only applies to a certain class of people which I will not get into. There are different classes of people with different ranges of ability to progress)
-d