In the 1830s, there were two notably developments in French army headgear, part of a gradual, very gradual, Europe-wide movement away from the uniforms of the Napoleonic period.
In 1830, the French invaded Algeria. As in colonial campaigns elsewhere, the heavy regulation chaco proved unsuitable for desert campaigning. Officers at first and then the troops soon adopted non-regulation, peaked cloth caps referred to generally as casquettes d'afrique. In 1839-40, an experimental unit of Chasseurs à Pied was formed, known variously as the ‘Chasseurs Vincennes' or ‘Chasseurs d'Orleans.' Inspired in part by African experience, instead of the regulation chaco and coatee, they wore more practical tunics and a lighter cap, or casquette-chaco.
Remarkably, the generals saw the merits of this more practical headgear over the traditional bonnet de police, with its hanging flamme, in use since the 17th century, and c. 1843-45, together with a lower, conical casquette, a blue bonnet de police à visière was authorised. It seems that German soldiers in the newly formed Légion Étrangére may have referred to this headgear as kappi whence comes the familiar term ‘képi.' In 1852, the famous red crowned bonnet de police à visière or képi was introduced.
In the victorious if costly ‘Napoleonic' campaigns of the 1850s, the familiar image of the Imperial troops in their red caps popularised this characteristic piece of French headgear, famously adopted by the US army. In actual fact, there are images of Austrian cavalry in soft crowned conical caps from the years immediately following the end of the Napoleonic wars.
Gradually, the formal regulation casquette of the French army reduced in size, becoming lower and softer (retaining its pompon) while the képi became more cyndrical and its peak flatter until, as often happens, following the Great War the undress képi became the formal headgear of the French army.