Let \(N\) and \(P\) be smooth manifolds, \(\dim(N)\geq \dim(P)\geq 2\), let \(f: N\to P\) be a smooth map, and let \(\theta_N\to N\) be a trivial line bundle. The author shows that if there exists a fiberwise surjective bundle map \(TN\oplus\theta_N\to TP\) covering \(f\) then \(f\) is homotopic to a map with only fold singularities. Furthermore, additional conditions for finding a homotopy to a map which folds only along spheres (of dimension \(p-1\)) are given.