Keeping an open, partially empty soda bottle bubbly longer
The fizz is dissolved carbon dioxide (or carbonic acid with which it is in equilibrium) which is put into the bottle under quite some pressure (around $\pu{120 kPa}$ at $\pu{4 ^\circ C}$ or $\pu{250 kPa}$ at $\pu{20 ^\circ C}$; from a first glance on the google results).
$$\ce{CO2 (g) + H2O (aq) &lt;=&gt; H2CO3 (aq)}$$

Because the bottle (in first approximation) is a closed system, there is an established equilibrium between the "air" in the bottle (gaseous $\ce{CO2}$) and the dissolved $\ce{CO2}$. Once the bottle is opened the equilibrium is disrupted and the dissolved carbon dioxide is no longer as soluble as before.

If you close the bottle again and use the little pump you essentially add pressure again, shifting the equilibrium back to where more dissolved carbon dioxide is stable. It won't be as effective as the factory produced bottle, but it will slow down loosing the fizz.

The volumes also play some role here, ideally the aqueous phase should be much larger than the gaseous phase, as you need more molecules producing the pressure if you have more space. An half empty bottle therefore is not ideal to start with.

Squeezing the air out of the bottle would in the worst case result in the opposite effect that you want, in the best case it won't change much. As you correctly remarked, the bottle reforms which means that you would have created lower pressure, essentially dragging out more carbon dioxide molecules. If that was not the case, you would reduce the volume of the gaseous phase, which would probably help a little. Without adding high pressure though it will still loose most of the fizz over time, until equilibrium is reestablished.

In principle there is not really very much you can do. Keep the bottles refrigerated (as carbon dioxide is more soluble at lower temperatures) and try to finish them quickly. Alternatively, get yourself a soda maker and enjoy all the fizzy drinks in all the customised quantities whenever you want. (I guess, this is not a plug for soda maker machines, I personally prefer non fizzy drinks.)