I am pretty convinced that this is an edition of the Parker Bros. game "Fox and Hounds" from 1948. Obviously, this one would have had a different name, since there is a hare instead of a fox.
However, I can't figure out what the name of your game might have been, and I haven't been able to find a matching board.
Still, hopefully this is a helpful clue.
Here is an image of the game board for Fox and Hounds:
It's a different board but it has the same red paths.
The rules are very simple. The hare is trying to reach the Hare's Burrow by exact count and the hounds are trying to land on the hare. Each player rolls a die on their turn and moves the corresponding number of spaces. Switching direction during a turn is not allowed. Only the hare is allowed to move along the red paths and, if on a red path, must move off it if possible.
The sign near the top of the board looks like it says "new cards", so maybe your version doesn't use dice?