A great place to start, is by demonstrating, what can be done with additional budget. Showing what sort of opportunities it can open up. But also then making a prediction on what sort of results that it will get for the business.
Capital is always constrained, and you have to fight to secure that capital, the more compelling the argument, the more it makes sense for additional investment. You kind of have to put yourself in the companies shoes, should I put more capital there, or somewhere else.
A platform to getting more budget, is making sure you have justified your existing budget, demonstrated its effectiveness, ROI and explained to non marketers the value it creates. If you don’t have these as your platform, asking for more budget will be challenging.