Unfortunately, it's not possible to automatically save multiple stages of the progressive render. The nearest you can get to this is to render multiple images with a small number of samples and then combine them to produce a result - similar to the progressive render.
To achieve this you need each 'frame' to be rendered with different 'seed' setting. The 'seed' is the source of the 'noise' in the render. Different seeds produce different noise and when these are combined it results in the noise effectively cancelling out. The seed for the render is available in the Sampling render properties. Alongside the Seed value is a clock icon - clicking this will use the frame number as the seed and this will result in a different seed being used per frame.
As an example, here are multiple frames rendered with 1 sample each, each with a different seed :
Combining multiple images is most easily achieved using tools outside Blender - such as ImageMagick. To combine multiple images using ImageMagick you can use the following command line :
magick convert *png -average result.png
This produces the following result :
To produce the 'progressive' result (multiple images at different stages) you would simply combine fewer images for each 'earlier' stage in the process (for example, images 1-20, images 1-40, images 1-60, etc. for progressively more detail).
Note, however, that better results will be produced from the 'actual' progressive render due to losses resulting in each individual frame - for example, values outside the 0.0 to 1.0 normal range are not stored in individual PNG renders. This will result in the loss of some details such as caustics.