When filling in low spots on a lawn, fill to a depth of no more than 1inch (25mm) at a time and fill in no more than twice a year, once in early to mid Spring and once in early Autumn (Fall).
As long as you fill in no more than 1inch (25mm) at a time, the grass will happily work its way through and fully establish itself at the slightly new higher elevation.
If you have a few inches (75mm or more) to fill in, you can most definitely remove the sod, fill underneath, then lay back the sod:
Carefully remove the sod.
- Remove in sections, something like 12inch (300mm) x 12inch (300mm) x 4 to 6inch (100 to 150mm) deep.
Add your fill material (see below for recommendation).
- If adding more than 4inch (100mm) depth of material, add in 4inch (100mm) layers, each layer should be "lightly" tamped down (do not over compact).
Carefully lay back your sod, the sod should sit approximately 1 inch (25mm) above your finished required level ie The relaid sod should be sitting high.
- Lay a scaffold board (or similar) over the relaid sod, shuffle up and down the board a few times. This will ensure good contact is made between the fill material and the sod, and the sod will lower (settle) to it's new final required level.
Sprinkle a thin layer (¼ to ½inch / 6.25 to 12.5mm) of compost over the area you've finished bringing up to grade.
Water once a day for the next 7 days, so the area remains moist (but not saturated).
- After that time, water your entire lawn as you would normally.
Fill material: Personally I would use a 50-50 mix of "high quality" screened (¼inch/6.25mm sieved) top soil & compost for filling in low spots in a lawn.
"High quality" = Make your own from compost and soil (materials) you control, or buy from a local independently owned garden nursery, or if you're lucky enough to have access to a local (free) community composting facility that is known to output good quality material.
- Basically you want to reduce the risk as much as possible of bringing in lots! of weed seeds into your environment.