You may be interested in this older post of mine.

One simple change I might make is to search for "social insurance" programs, vs welfare programs— social insurance being common terminology. Social relief and social assistance would probably also turn up a lot.

You'd probably be interested in the 1834 New Poor Law in Britain, which created a system of locally-run workhouses to "deal with" and reform the poor. The Poor Law was on the books until 1948, when it was repealed and replaced with the National Assistance Board. Anything relating to workhouses, the Poor Law, etc., will come up with a lot. There was at least one Poor Law Commission to investigate the implementation of the Poor Laws and the state of the workhouses that you could look into.

A lot of today's welfare bureaucracy has its roots in the private voluntary organizations of middle-class women, which were designed to assist working-class women and their children. This is especially the case in the US and Britain, two countries with powerful female reform movements and governments that were much stingier about providing maternal and child welfare benefits (thus requiring female reformers to step in). You can read much more about this, and find examples from across the US and Europe, in Mothers of a New World: Maternalist Politics and the Origins of Welfare States, eds. Seth Koven & Sonya Michel, Routledge, 1993. (The editors also published an article with a similar title in the American Historical Review, 1990.)

I'd also recommend searching maternalism (particularly if searching the secondary literature) but also things like child welfare, day nurseries, maternity leaves, mother's/widows pensions, the health and well-being of children, motherhood/mothercraft, and infant care/ infant mortality— it was around questions of mothers and children that states first considered and implemented relief programs for the poor.