Here are some hacker options:
bridges <- df[df$month == 'Jan',]
bridges$year <- bridges$year - 1
bridges$month <- NA
ggplot(rbind(df, bridges), aes(month, value, group = year)) +
geom_line() +
scale_x_discrete(expand = c(0,0), breaks = month.abb) +
coord_polar()

Obviously adding extra data points is not ideal, so perhaps a more elegant answer exists.
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