I have worked for charities for over 20 years and managed quite a few employability projects for people who are disadvantaged/vulnerable, had mental health issues, a history of addiction or a criminal record. So basically people who found it harder to get employment.
We usually had training courses where people could access free IT and ESOL (to improve their English) training as well as training in things like catering, child minding, gardening and so on and gain basic qualifications.
We also helped them get work experience through volunteering or work placements, often at the community centres where these projects were delivered helping at reception, the kitchen/cafe if there was one or with admin tasks. We also worked with local adult education providers to find them other courses if needed.
This meant they had positive things to put on their CV and we could also provide them with a reference for employers.
I would advise you to look at what similar projects are run by local charities, community centres or the council that could help your partner improve his skills, get some experience and build up his CV/references now that has sorted is immigration status.
As other people have already said the easiest jobs to get are usually hospitality/catering, construction, removal/delivery, cleaning jobs.