Warriors get started on Saturday night with a tough visit to Belfast to take on Ulster.
The club's second league triumph has raised expectations they can challenge at the top again this season and make a mark in Europe in the Champions Cup.
"That is the expectation I'm trying to keep out of our environment," Smith told the BBC Scotland Rugby Podcast.
"If we're going to start pinpointing games and competitions, we're going to miss the trick. We've got six URC games now, then there's four Test matches, then there's three URC games, of which two are the derbies, and two European Cup games until Christmas.
"That's 15 very, very tough games to be involved in all of them. So now it's about what can we do with a second group of players that is now making the step up into this environment.
"If we're going to start getting ahead of ourselves, injuries will happen and it will then suddenly put us off track.
"I can promise that everybody here wants to win every game, every competition we play in, and that is what we strive to. But we'll have to take it one game at a time, see what we have available, manage the players best we can.
"Hopefully by doing that, we will just keep that expectation at bay and the unnecessary pressure outside of our environment."