Is it possible to use the power from the headphone jack line out to
amplify the signal in order to get the speaker to be louder?
Here's the thing: The power in the headphone jack is in the audio itself. The power in a normal wall outlet is 50/60Hz AC. The power in a headphone jack is also AC and in the frequencies that you want for your audio! There is no additional power contained in a headphone jack other than the audio itself.
So, if you could somehow convert the power in the headphone jack into something useful and convert it back into a form for driving a speaker, and could do it with 100% efficiency then you would end up with exactly what came out of the headphone jack in the first place.
In short: You can't amplify the power without another source of power. But all is not lost!
What you really want is louder audio, and a way to do that is to make your speaker more efficient. Speaker efficiency is usually expressed as "sensitivity" and is specified as XX dB at 1 meter at 1 watt. A normal speaker is 75 to 90 dB (@ 1 meter @ 1 watt). It might not look like it, but 75-90 dB is a huge range! You can simply get or build a speaker with better efficiency.
Another thing that effects efficiency is the speaker enclosure itself, and the type of speaker. Look into horn speakers. There are horn speakers that have sensitivities up to 105 dB (@1m@1w). Although these will likely be too large for your application, it might be interesting anyway.
Driving an 8 ohm speaker with a headphone jack is questionable. Some devices will do this just fine. Some devices might distort but otherwise be fine. And other devices might be permanently damaged. Unfortunately there is no good way to find out without just trying it.
In theory you could make an impedance matching transformer that would show a proper load to the headphone amplifier, but who's output is good enough to drive an 8 ohm speaker. Just keep in mind that a transformer will not increase the power, only convert the form of that power so different devices can play nice together. I have not seen such a transformer for sale, but I also have not looked for one. But in theory you could make one.
Given all the different issues, if it were me I would figure out a way to make a small battery pack with an amplifier in it. Yes, it's bigger than what you want. But it will probably be the best compromise between volume, size, cost, and robustness.
It's hard to tell what you are asking because some things you say don't make sense. In the schematic you show, both outputs are clearly referenced to ground. Pin 5 produces the inverted signal and pin 8 the non-inverted, but both are referenced to ground. The DC offset of both will be close to half the supply voltage.
However, if the point of this block is to be a preamp, then the choice of this part makes no sense. You didn't supply a link to the datasheet so I don't know what a LM4871 is exactly, but from the schematic you show it appears to be a small power amp. You can see that the gain of each output is 1 referenced back to the input. The main job of this circuit seems to be to provide power gain (lower impedance output) as apposed to voltage gain, and to produce a differential signal to drive a small speaker with directly. This does actually give you a voltage gain of 2 accross the speaker relative to the input, but the main point is to be able to drive the speaker with ± the supply voltage. That is useful to put more power into the speaker from a low supply voltage.
If you want a preamp, make a preamp, and use parts intended for some voltage gain with good characteristics of noise, linearity, and distortion. This is not what the LM4871 is intended for, given the schematic you provided.
Best Answer
Each channel output is a H bridge therefore you cannot cross connect speaker wires to form a common 0V. You could use a transformer to convert a balanced output (H bridge output) to a single ended output and you'd need one transformer per output.
You might be able to use the half-H bridge outputs but distortion and other problems may occur due to imbalanced loading. Read the data sheet to see what it says.