It is my understanding that stars the size of our sun will go through two red giant phases. The first one will occur when fusion of hydrogen to helium begins to happen in a shell surrounding the collapsing helium core. Then, as temperature and pressure increases in the core, the so called helium flash will occur when fusion of helium to carbon begins (the triple-alpha process). This will lessen the "pressure" in the core, and the star will shrink again. Later, as the star begins to run out of helium, and the carbon core begins to collapse, a second red giant phase will occur.
My question is as follows: Stars between roughly 2.2 - 8 times the mass of our sun will, as far as I understand, initiate the helium-carbon fusion without any helium flash. Does this mean that these stars will not shrink in the same manner as our sun will between the two red giant phases? After all, we will not have the same sudden release of "pressure" in the core here, so will these stars just remain red giants? Or will there be a gradual shrinking of these stars too in the stage before the carbon core begins to collapse?
If someone can explain this to me, or correct any misunderstandings I might have, then I would greatly appreciate it!