Your husband probably knows this, but Caterpillar have a business in France, based in Grenoble, which advertises jobs via Pôle Emploi (the French National Job Centre) :
Caterpillar France Recruitment
Perhaps there are internal avenues that might be explored via HR within Caterpillar itself, unless of course, he no longer wishes to work for them.
As an engineer, he would likely be considered for a job as a “cadre” - this is a form of socio-economic status which is both a help and a hindrance - trying to apply for a less qualified post will be incredibly difficult, and to the converse, looking for jobs as a “cadre” will reduce the number of jobs available, but if your husband doesn’t speak French will be even more difficult (not impossible, just difficult).
By law, job vacancies within most (all?) French firms have to be posted with the national job centre system at least for a while, but as I have found in my own personal experience, when I was briefly unemployed before setting up my own business, are essentially useless. It is a system where one, as an individual, is confined to a box with a specific “label” - the job centre will only call you or contact you when they have a job offer that exactly matches the “label” they have determined for you…
Your husband might be better off searching for a job on a specialized website, such as the APEC (link), or CadrEmploi (link)
Alternatively, services like Monster recruitment or LinkedIn Jobs Search might bring up something.
Unless the job posting is with an international company specifically looking for an English speaker, expect the interview and all dealings with HR or a recruiting agency to be in French. Even though many French mechanical engineering firms can and do use English as a medium for international business dealings, the common language used internally will still be French (this is actually the law in France, despite some firms having attempted to impose all-English communication).
Good luck with your searching !