S.H.
Well, my kids, have always woken up at about 6-6:30am. Since they were babies.
No matter what. No matter what time they went to bed, no matter if they nap or not. It is just their inner clock.
Don't know what you can do... sorry.
But if your 5 month old is not bothered by it, nor wakes up because of it... then, that is good?