The “escape velocity” is the velocity needed to escape the gravitational field, which is still strong even after the end of the atmosphere layer, it actually goes infinitely far. And yes, you can go slower, this is the concept behind the idea of the space elevator.
To escape a gravitational field you need an amount of energy that depends on how deep in the field you are. In theory, it doesn’t depend on the path you take or the initial speed. That energy, expressed as kinetic energy, corresponds to a speed (the masses cancel out).
When talking about rockets, this means that a rocket engine must contain at least enough energy to reach that speed if it were in free space, otherwise it wouldn’t be able to escape earth. That energy can be released quickly, like in an actual rocket, or slowly, like in a space elevator.
Since atmospheric drag and rocket technology plays a role here, it’s more efficient to quickly release most of the energy at the beginning instead of going up slowly.
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