In cases like that,and i see it day in ,day out, in the shop,most most likely it's the ALTERNATOR belt that needs tightening;
The reason it quiets down afterwards,is because the alternator already replenished the battery charge that was spent by the starter,and also,it's warming up,getting a better grip; stil..it NEEDS tightening,(renew it,if it is very old!)it will only get worse;
There is an easier way to test ,to make sure: get a garden hose,(or just a pitch with water)and totally wet the belt and its pulleys as it is turning(engine idling): the water will act as a lubricant,and the slipping belt(if it's the case) will completely stop squealing; the water dries in seconds and all goes back to what it was like before again,but you got your answer.