As you said, evaporation occurs at all temperatures, including those lower than the boiling point.
Evaporation occurs (1) at the surface of the liquid, its interface with the atmosphere, (2) into minute cavities in the inside wall and bottom of the container, below the surface of the liquid. These cavities, not usually noticeable, contain air, but this is soon joined by saturated vapour. Suppose that the liquid is continuously heated. When the combined pressure of air and saturated vapour (SV) in a cavity just exceeds atmospheric pressure (pressing down on the liquid/air interface), the air and SV in the cavity expands forming a bubble that breaks away from the container wall and rises to the surface. The cavity now contains mostly SV and little air. When the SV pressure is just greater than atmospheric, the SV expands forming bubbles that are able to escape through the liquid to the surface. The cavities keep refilling with SV (they never completely empty), bubbles of SV rise and so on. This is boiling !
Perhaps I need to emphasise the point that evaporation into cavities doesn't result in immediate escape of vapour to the atmosphere, because the surrounding liquid, with its closely packed molecules, acts as a seal. Escape is only possible when the pressure in the bubble is just greater than atmospheric pressure. Only then can the bubble expand, detach itself and rise to the surface.
re your question: "So, can I assume that just before it starts boiling (in saturated water), the entire 1 atmospheric pressure is only due to the part of water which is evaporated before?" No, you shouldn't assume this. The evaporated liquid (now vapo[u]r) has virtually no effect on atmospheric pressure. I wrote the above answer because it looked to me as if you were very confused about the whole idea of boiling.