There are multiple ways to do that.

You can edit the ggplot object

gg_figure <- DimPlot(combined, reduction = "umap", split.by = "orig.ident", label = T)
gg_figure$data$orig.ident <- gsub("treat","treat_day1",gg_figure$data$orig.ident)
gg_figure$data$orig.ident <- gsub("control","control_day1",gg_figure$data$orig.ident)

You can change the name of the groups in the object before plotting

combined@meta.data$group <- gsub("treat",...)
# or
combined@meta.data$modified_group <- combined@meta.data$group
combined@meta.data$modified_group <- gsub("treat",...)

Plot each group individually and put them together

# split combined into 2 objects and then 
p1 <- DimPlot(treat, ...) + ggtitle("trat day1)
p2 <- DimPlot(control, ...) + ggtitle("control day1)
p1 | p2

A simple trick would be to add an additional column in the metadata slot of the Seurat object. Something like this:

combined@meta.data$group <- ifelse(combined@meta.data$orig.ident == "treat","treat day1","control day1")

and then rerun a Dimplot

DimPlot(combined, reduction = "umap", split.by = "group", label = T)