Melatonin is what helps you to go to sleep and stay asleep. DHEA levels build up during the night to get you started for the day. Melatonin keeps you from waking up too soon as DHEA levels start to rise. I think you could consider adding DHEA, taking it when you wake up.

Melatonin doesn't tend to have a taste to it. You could crush the pills and mix them into something. Peppermint or chamomile tea, for example. That by itself could help it to kick in faster and be more completely metabolized by morning.