The sleep icon (Zzz) is a heads-up that your tracker hasn't picked up any movement in the last 15 minutes, suggesting that your pet is probably resting at this time.
As soon as the tracker detects movement again, it will update its position as normal.
You can still start LIVE Tracking whenever you need it, but keep in mind that LIVE Tracking works best when your tracker is outdoors with a clear line of sight to the sky.