When you have changed all the strings, tweak the saddle into the upright position, tension the strings a bit further and pull the saddle back again. Keep repeating the process until there is too much string tension for you to be able to move the saddle easily. By this stage the saddle should be retaining its vertical orientation in any case. I don't recommend trying to tweak the saddle backwards under full string tension. However, for the amateur there is too much risk of the whole bridge slipping and causing damage to the top of the instrument.
Many players recommend the Brekke two-piece bridge .
This has a unique design which lessens the tilting effect. Strings Modern strings are a miracle of technology. They are created to exacting specifications, including precise diameters. A brand new string of good quality should have uniform diameter, winding and plating thickness along its length. Change any of these factors and nasty overtones will, not surprisingly, creep in.