Warm the needle up to about 105 degrees, either with hot water at the sink or use a heater, flame, or sun light. (A cold piece of ice or steel can be a bit painful to the touch, without penetration).
What you are injecting, could also use some warming up, especially if it has been in the refrigerator. (Even your outer layer of skin wouldn't appreciate being exposed to a 33 degree temperature liquid).
Why 105 degrees instead of 99 degrees, approximate body temperature?
By the time the needle is inserted, an air conditioned room will bring the temperature of the needle down a little.
The body stance, of the shot giver, should be very stern. A wobbling inserted needle could cause a pain of its own.
Dentist, administering a numbing shot that requires a lot of numbing, can at first, just barely prick the gums, deaden them slightly, then withdraw the needle. Give the gums a little time to get numb, then proceed with the rest of the injection.