Not exactly sure what you mean, but I may have some?
When looking for a mate, I like to first check all the squares the king currently has available, and think of what combination of pieces (or single piece) and where they would be to cover all these squares.
You mentioned the pattern of squares knights cover. Another mating thing that may be basic but I use it is to know what the pattern of free squares looks like for example when the queen checks from an orthogonal adjacent square (there are at most two escape squares, diagonally backwards). When checking diagonally adjacent there are at most two escape squares orthogonally away. Sometimes this helps in calculating the end of a mating attack, I see certain squares are covered then look for a way to put a queen on the square that will give mate.
The rook also. When "touching" the king while giving check it gives a certain pattern of 4 escape squares.
A knight, a pawn, or a king three squares away from a knight impedes it's movement. Some people decide to remember that if a piece is on an opposite color of the knight it can't be attacked next move (so knight gives check from light square, any dark square you move to can't be attacked next turn). But there is a square the same color that makes you safe for minimum of 3 moves, namely this is two squares away diagonally. This can also be a shortcut to see if you can attack something quickly or get to an outpost. If two squares away diagonally you can pretty much forget about it (takes 4 moves to land on the square).
Oh, and because the knight switches square color each move, if you're trying to figure out how fast you can get somewhere it's always an odd or even number. e.g. if you can't in 3 then it's probably 5. If you can't in 2 then it's probably 4 (handy if needing to race across the board to a certain square).
If I'm in a hurry and I'm trying to calculate a position involving a pawn race I'll sometimes use the color and sometimes the rank number (so I'm not even shifting my eyes to the enemy pawn). e.g. I imagine my knight to d6 then think "5" I imagine my next move "4" next move "3" and I know when I say "1" that the enemy pawn is on the 1st rank and is a queen. Or I'll say to myself "white" then "black" with the same idea. I'll also use this for king moves (both kings are making their way into the center). I'll give myself 3 moves then see all the different 3 moves paths the enemy king may choose to take for example. When I find his ideal path I'll go back to alternating turns.
Other tricks aren't so explicit, and almost subliminal. e.g. a pinned piece isn't regarded as defending something, and in treating whatever it is as undefended you see a possible fork (just for example). Or a piece is the sole defender of two points. You attack one point and think of it as having already removed a defender of the other point.
Oh, one more trick. In pawn endgames when you're fighting for tempo (maybe kings are in mutual zugzwang) then you don't want to attack an enemy pawn with your pawn when they have the option to either capture or push. A capture forcing a recapture gives the move to them. A push gives the move back to you, so either way they'll win. This can greatly simplify how far you have to calculate when you regard that as something you can't do (and if they do it you win).
Post Calculation Tips
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I thought it would be helpful to have a thread where we share our tips for calculating in chess. Ones that I know and use:
In the endgame you can quickly see if a king will reach a pawn before it promotes. Draw a diagonal line from the pawn to the back rank. Create a square with the ends of that diagonal being opposite corners of the square. If the King can step inside that square, then the King can capture the pawn. Obviously there can be exceptions when there are other pieces involved or when the pawn is still on the second rank.
When two knights are right next to each other, it's tough to keep track of all the squares that are covered. It helps me to think of the covered squares as 4 tetris pieces. So for instance, if the knights are on d4 and e4, then the squares e6 f6 f5 g5 make up one of the tetris pieces.
When multiple minor pieces are coordinating an attack against a king, it’s tough keeping track of which squares are being covered around the king. I go through each piece to see what squares it is covering and then I forget about the piece, I only focus on remembering the covered squares. So for instance I’ll keep track that e7 is covered, but won’t think about which piece is covering it. Then when I go to calculate my candidate moves I check to see how that changes which squares are covered.