Can you still use a curling iron on curly hair to get bigger waves, without straightening it first?

If you have tight curls, I don't think that would work because the curl is already smaller than the barrel of the curling iron. You could blow-dry your hair into messy waves and then polish them up with the curling iron instead of fully straightening them out and then curling them again. Or use big rollers on your hair when it's damp and blow-dry them.

Can you still use a curling iron on curly hair to get bigger waves, without straightening it first?

totally! i have very curly hair, and sometimes i use a 3/4 inch curler to touch up when im going out. but if you use like a 1 1/4 inch curling iron it will mostly flatten out your hair, and give you bigger, wavier curls, the only down side is that because it flattens out your hair, it may not hold as strongly as normal curls, it may start to hang and straighten. but it will look hot for a night out!