Sure, you can and should charge for added work you do or extra hours you spend meeting a tight deadline. You might even charge at a premium rate, but that’s not what really pays.
What pays is doing a great job for someone when they really needed it and knowing… they’ll be back.
When you go above and beyond to complete a project, you’re earning a repeat client. So while the added revenue from that project is nice, it’s all the future revenue you’re securing for your business that’s the real reward.