What is the Relation Between Fat, Scar Tissue, and Collagen?

You describe what might be a hypertrophic scar from the prior procedure. This type of scar should not improve as a result of further liposuction, and in fact, could be stimulated by the procedure possibly and get worse. Scars are made of collagen, a protein that you manufacture. Fat cells are not related to collagen. Any procedure can induce collagen, though, and this helps the fat cells that remain contract together to some degree. Some people heal with excessive collagen, and the insertion scars, through which the liposuction cannulae are introduced, can develop into raised scars.

Scar is predominantly composed of collagen, a protein. This is a relatively inactive tissue which does not dramatically change once stable
Fat is a tissue which is composed of adipocytes. It is very active and can respond to weight gain and loss.  

 First, if you just had a baby I would not have any surgery performed. You should wait at least a year.  The cause of stretch marks is unknown and probably genetic in nature.

Fat is a living tissue made up of cells specialized to store unused calories as fat.
All injuries heal by deposition of scar tissue, a biological glue which is constantly modified over a 8 - 12 month period after the injury / operation. But as stretch marks demonstrate it is permanent. You cannot permanently erase scars or remove it.
Collagen is but a component of scar tissue. It is produced by and deposited by scar cells (Fibroblasts) found in nearly all tissues.
When we say that something "produces collagen" it is a clever marketing way of saying the product causes an injury resulting in scar deposition.
Obviously the kicker is HOW MUCH ? A little scar tissue UNDER (not in) the skin is not important and May even shrink the skin for a while, on the other hand a large injury (gunshot, scalpel, burn (Laser, Acid, or hot water) - ALL produce a LOT OF COLLAGEN. But, not exactly where we want it and how we want it.

Fat in the human body is referring to specialized cells that accumulate and store fat as energy for future use. It is fairly soft and is what is removed with liposuctioning.
Scar tissue is what the body makes to heal injuries. It does have collagen as part of it. Scar tissue will be found in any area of injury including liposuctioned areas.
Collagen is a protein that is found naturally in the body in many areas inclusing aroud fat and in scar tissue.

Fat is composed of lipids and fatty acids.
Collagen is a protein, and there are 29 forms in the body.
Scars contain Collagen Type I which makes up about 30 % of all protein in the body.
Liposuction will cause minute scar tissue which helps contract and hold the results. Laser purports to further help in this regard. Some people just don't get stretch marks at all. I do not know of any studies suggesting liposuction preventing stretchmarks of later rapid weight gain, pregnancy, or bodybuilding which occur after liposuction. Some people just dont get them no matter what stress is imposed.
Stretch marks are caused by the skin stress imparted by rapid weight gain and increased water in the tissues and in the present of certain steroid hormones, as in pregnancy, and anabolic agents used by bodybuilders. The skin will tear with disruption of small blood vessels imparting the red color. The hormones can introduce water that puts pressure on the collagen attachments causing them to pull apart.
Reduction of stretch marks by laser is extremely limited to selected cases, and mainly for red ones; (which can improve 50 % on their own) as the color fades. The laser reduces swelling, blood vessels, and promotes contraction. It does not erase them but only fades the discolored ones from purple and red to pale tones.
The pale marks do not generally respond. There are scattered reports of Retin A and other topical agents for light stretch marks causing major fading over years. Micro Retin A may be prescribed by a number of practitioners as it is more easily tolerated for limited areas of these marks.