We study cosmological properties of type IIA compactifications on orientifolds of SU(3)-structure manifolds with non-vanishing geometric flux. These compactifications give rise to effective 4D N = 1 supergravity theories that do not fall under some recently-proven no-go theorems against de Sitter vacua and slow-roll inflation. Focusing on a well-understood class of models based on coset spaces, however, we can use a refined no-go theorem that rules out de Sitter vacua and slow-roll inflation in all but one case. The refined no-go theorem uses the dilaton and a specific linear combination of the Kähler moduli, which is different from the overall volume modulus. It puts a lower bound on the first slow-roll parameter: ε ≥ 2. The only case not ruled out is the manifold SU(2) × SU(2), for which we indeed find critical points with ε numerically zero. However, all the points we could find have a tachyon corresponding to an eta-parameter η ≲ −2.4.