The audio fidelity of those things must be terrible! There are all kinds of effects that are going to roll off the high frequencies on you. I guess you have to equalize the heck out of the signal to compensate.
Anyway, to get more sound, you need to increase the motion of the magnets. To increase the motion, you need a stronger field. The strength of the field is going to depend on both the construction of the coil and the capabilities of the driver.
Addressing the coil first — it may seem counterintuitive, but if you're driving the coil with a voltage source, and ignoring resistive losses for the moment, you want fewer turns in the coil rather than more. The issue is that the field strength is directly proportional to the number of turns for a given current. However, the inductance is proportional to the square of the number of turns, and the current is inversely proportional to the inductance when the voltage is held constant. This means that the total field is inversely proportional to the number of turns — at least until you hit the maximum current that the driver can supply.
Now let's look at the resistance of the coil, which is also directly proportional to the number of turns. If the resistance is significantly less than the inductive impedance, you can ignore it. Let's look at some numbers. You didn't specify the gauge of your wire, so I'll assume AWG30 as a starting point. The coil has to be big enough to slip over your head, so the circumference needs to be something over 2 feet — let's call it 70 cm. Therefore, with 70 turns, you have a total resistance of
$$330 \frac{m\Omega}{m} \cdot 0.7 \frac{m}{turn} \cdot 70 turns = 16 \Omega$$
The inductance is going to be on the order of
$$\frac{a^2 n^2}{9a+10b} = \frac{4.5^2\cdot 70^2}{9\cdot 4.5 + 10\cdot 0.25} = 2.3 mH $$
This will have an impedance of about 140 Ω @ 10 kHz, 14 Ω @ 1000 Hz and 1.4 Ω @ 100 Hz. Clearly, the inductive impedance dominates over 1000 Hz, and the resistance dominates at lower frequencies. So at lower frequencies, a constant-voltage source will drive a constant current. In this regime of operation, changing the number of turns has no effect, since any increase in the field strength due to having more turns is directly offset by the corresponding decrease in current.
Therefore, it makes a lot more sense to focus on your driver circuit. An LM386 is going to be rather limited in this kind of application. For several reasons, it would be far better to drive the coil with a current source rather than a voltage source. Most of the inductive and resistive effects could then be ignored, and you could directly control the volume by controlling the current. However, such a driver would require rather high voltages, especially at higher frequencies.
You might take a look at the Howland current source, which can drive moderately high currents with low overhead.
