I can't say this for certain, but I believe the combination of length of day and elevation of the sun at the zenith is a unique combination for each day of the year and latitude. So I think you can get your latitude the same day. But I'm not sure you don't have the same two day ambiguity. Otherwise I think the combination is unique. Even the North-South issue can be resolved because of the eccentricity of the Earth's orbit making things a little different in the two hemispheres. But you may also be foiled beyond the artic/anartic circles where the sun never sets. Then you only get one parameter, the elevation at the zenith. But you might be able to make up for that by measuring the time between the sun at due east and due west... other than at the poles where there is no east or west... ;^)
Rick