On Call Sleep Question

If you are on call and you receive a call at say 3:45 am and you resolve the issue by 4:30 am. Is it then worth trying to go back to sleep to wake up for work the next day or should you just stay awake and power through it?

I'm asking because this happened to me and I went back to bed, did not feel tired at all and when I eventually fell asleep I got maybe an hour of extra sleep and I felt like complete garbage when my alarm went off and pretty much like that for the remainder of the day. Whereas I feel like if I just stayed awake for the extra time after 4:30 am I might have not felt as bad?

What are your opinions on this?

Edit: I'm appreciating all the responses and taking the information in. Sounds like this is not a clear cut case that is a simple yes do this or no don't do that.