Yup, any beer remaining in the spigot, exposed to air will get infected. So you need to clean/rinse/spray that out thoroughly and sanitize it after use. Then resanitize right before you take your next sample, bottle from it, or use it to transfer to a bottling bucket.Just remember to spray starsan or something up the tap after using it, all kinds of nasties may grow there otherwise.
If you don't... you may get infected beer.