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In geometry, understanding the topologies and the differentiable structures of manifolds in constructive ways is fundamental and important. It is in general difficult, especially for higher dimensional manifolds. The author is interested in this and trying to understand manifolds via construction of explicit fold maps: differentiable maps locally represented as product maps of Morse functions and identity maps on open balls. Fold maps have been fundamental and useful in investigating the manifolds by observing (the sets of) singular points and values and preimages as Thom and Whitney's pioneering studies and recent studies of Kobayashi, Saeki, Sakuma, and so on, show. Here, construction of explicit fold maps on explicit manifolds is difficult. The author constructed several explicit families of fold maps and investigated the manifolds admitting the maps. Main fundamental methods are surgery operations (bubbling operations), the author recently introduced motivated by Kobayashi and Saeki's studies such as operations to deform generic differentiable maps whose codimensions are negative into the plane preserving the differentiable structure of the manifold in 1996 and so on. We remove a neighborhood of a (an immersed) submanifold consisting of regular values in the target space, attach a new map and obtain a new fold map such that the number of connected components of the set consisting of singular points increases. In this paper, we investigate cases where the numbers increase by two and obtain cases of a new type.