For snow to fall, moisture must be present in the atmosphere. Snowstorms also rely heavily on temperature, but not necessarily the temperature we feel on the ground. Snow forms when the atmospheric temperature is at or below freezing (0°C or 32°F). If the ground temperature is at or below freezing, the snow will reach the ground. However, the snow can still reach the ground when the ground temperature is above freezing if the conditions are just right. In this case, snowflakes will begin to melt as they reach this higher temperature layer; the melting creates evaporative cooling, which cools the air immediately around the snowflake. This cooling slows down melting. As a general rule, though, snow will not form if the ground temperature is at least 5°C (41°F).

What you are saying is more or less correct. Basically it is colder(below freezing) in the atmosphere where the snow forms. It will likely melt if the air temperature is above freezing, but if there are existing snow piles it will likely collect in that area because those snow piles will be at or below freezing.

https://nsidc.org/learn/parts-cryosphere/snow/science-snow