If you're applying for the banks and big 4, you've got very little chance without any work experience.
If you're applying for anywhere else, you've got very little chance with nothing to separate yourself from everyone else.
As you say, internships are mainly for undergraduates rather than graduates, so what I would advise doing is showing the employer you're willing to specialise in an area and give yourself an edge over all the other candidates with a Maths degree.
The way to do this would be to save up to do a Masters. I don't know if there's specific roles you'd like to go into, but it may be worth doing a Financial Mathematics masters to improve your programming, apply your numeracy in real-world based projects and give yourself a different experience to be able to refer to in an interview setting.
You can then start to look at pretty much any graduate role with either a problem solving, numerical or computational aspect to it and get your foot in the door.
Everyone has a Bachelors these days, but not everyone has a Masters.
(Source: got masters, got a job)