A month can make a big difference. Villagers can move away, flowers will die (important for hybrids), weeds will sprout everywhere. I don't know that there's a "safe" way to time travel. Going forward day by day is safer, at least your turnips don't rot. And going back in time dosen't always reverse any damage.
The safest way is to find out which villager is moving, stop them, save and quit, and then TT a month forward. Make sure you have the Beautiful Ordinance activated (actually activated; if not, get it set up, and then TT a day forward for it to start working).
Trust me, this definitely works. I've TTed a month and little more than than before in my game and haven't lost anyone yet. *knock on wood*
As long as you stop someone from moving first before TTing forward, you'll be fine. But in between the times you TT, make sure you are stopping someone from moving before moving forward again. If no one is asking to leave and you plan on TTing pretty far, TT one day at a time until you hear rumors of someone thinking of moving or if a villager pings you to move.