Over night diapers leaking

I have a 4 month old. Shes only just started sleeping through the night (Yay!!)

Before, she would wake up a few times and wouldnt go back to sleep for a while, so I would change her diaper a few times during the night.

Now shes sleeping through, I dont disturb her at all and leave her till the morning when she wakes up to change her. But every morning the diaper has leaked urine quite a bit and her clothing and cot sheets are wet. How do you get round this? Does it mean the diapers are too small? They fit her fine round the waist and legs. Do i have to wake her up half way through the night to change her?