Would it hurt my water pump if installed a second water pump on the output of the first?
If the problem is undersized pipe, you need larger pipe, or some other coping mechanism such as a large pressure tank at the house.

Placing a pump on the end of the small pipe will likely cause the second pump to cavitate, which generally damages the pump. Water can only be sucked so hard before it boils, at any temperature.

If you want a pump on the house side, you can do that, but you'd need to provide a large cistern to serve as the water source for that pump, so it's drawing from the stored water in the cistern, not the too-small pipe.

A larger pipe is a much simpler solution to this problem in most cases.