## Make data easily reproducible
df <- data.frame(day=c(24, 21, 20, 10, 20, 20, 10, 15),
             month = c("Jun", "Mar", "Jan", "Dec", "Jun", "Jan", "Dec", "Dec"))

## Convert each month-day combo to its corresponding "julian date"
datestring <- paste("1963", match(df[[2]], month.abb), df[[1]], sep = "-")
date <- strptime(datestring, format = "%Y-%m-%d")
julian <- as.integer(strftime(date, format = "%j"))

## Transitions between years occur wherever julian date increases between
## two observations
df$year <- 1963 - cumsum(diff(c(julian[2], julian))>0)

df$year <- 1963 + cumsum(c(0L, diff(100L*as.integer(
  factor(df$month, levels = month.abb)) + df$day) < 0))
df

day month year
1  24   Jun 1963
2  21   Mar 1964
3  20   Jan 1965
4  10   Dec 1965
5  20   Jun 1966
6  20   Jan 1967
7  10   Dec 1967
8  15   Dec 1967