You might want to look into NMF, which, unlike PCA/SVD, doesn’t aim to create an orthogonal projection. It works well for interpretability because its components cannot cancel each other out, which makes its features more intuitive to reason about. I think it is essentially what you want, although I don’t think it will allow you to find directly the ‘larger set of almost orthogonal vectors’ you’re looking for.
You might want to look into NMF, which, unlike PCA/SVD, doesn’t aim to create an orthogonal projection. It works well for interpretability because its components cannot cancel each other out, which makes its features more intuitive to reason about. I think it is essentially what you want, although I don’t think it will allow you to find directly the ‘larger set of almost orthogonal vectors’ you’re looking for.