Every h-cobordism embeds into $M times I$?

Suppose we are in high dimension. By the s-cobordism theorem, any h-cobordism has an inverse given by a h-cobordism with the inverse torsion. Thus we have a isomorphism from the composite to the trivial h-cobordism. Thus by, restriction we can embed our h-cobordism into $M times I$.

Is there a simple handle manipulation argument that doesn’t use the full force of the s-cobordism theorem? Does it work in low dimensions?