Unless you really want the narrowest possible tire, I would also suggest 27 x 1 1/8" as a good bet since you had been using 27 x 1 1/4" in a different brand.
Going to 27 x 1" may be too narrow for some rims... but if the wheels on your bike are what came with it, based on the close clearances, I think the rims might be OK for narrow tires.