Adding an additional light to a 3-way swtch

I'm going to do my best to describe this without pictures because I'm hoping they won't be necessary.



I've got some stairs going down to the basement. Currently, there are two wall-mounted stair lights on the same 3-way switch. One switch is at the top of the stairs, and the other is at the bottom, obviously.



The problem is that these stair lights are pretty worthless in terms of illumination. I have a landing half-way down the stairs, but the builder never thought to put a recessed can light or wall sconce there. So, I want to add a wall sconce at the landing, and wire it into the same 3-way switch as the two stair lights.



One of the two stair lights is easily accessible via my basement utility room. So, rather than run my wall sconce wiring direct to one of the switches, I thought the easiest thing would be to run it straight down into the utility room and tie it in off the stair light. Can I do this?

Adding an additional light to a 3-way swtch