Every time a string is fretted it is also stretched. To touch the strings means always to increase their tension. And as you stretch the string it will rise in pitch for a few cent!
That´s the problem and here´s the solution.
With a lower tension of the string than you normally do, you can get in pefect tune the 1st fret.
Once in tune the 1st fret (F;B;Dis;gis;c;f) , you can see on the tuner two exact values:
1. How much out of tune the 12th fret.
2. How much out of tune the open string.
The first you can compensate on the saddle, the second on the nut.