There are no other reasonably direct methods, but there are definitely indirect methods. One, in @Alex Hajnal's answer, the higher CMB temperatures measured further out is a very nice indirect measure.
Another indirect piece of evidence, which no one has noted yet, is that as we look further and further out, the universe looks younger and younger, and less and less like what we see in our neighborhood. You are pretty much forced to explain that scientifically by saying that the universe had a beginning on the rough order of 10 billion years ago, and that stars and galaxies only started forming then. (This isn't proof of a Big Bang specifically, but it does eliminate most alternatives to it. The Steady State model, for example, is falsified.) It's very very hard to explain what we see except as being due to a universe expanding from a hot dense state ca. 1010 years ago.
More indirect evidence comes from General Relativity, a theory of space, time and gravity which is very well verified -- it's been tested for a century now and challenged by countless other theories, and only GR has passed all experimental tests. GR robustly predicts that a static universe is impossible and that it must either be expanding or contracting. This is indirect evidence from mostly local experiments.
Yet more indirect evidence comes from nucelosynthesis calculations which show that the H/He/Li ratios that we observe in the oldest and least evolved stars is exactly what we predict based on applying the measured properties of nuclei to a Big Ban fireball.
There's so much science other than the red shifts which point to the universe expanding from an initial very hot, dense state that even without the observation of red shifts, we'd eventually be forced to that conclusion.