Hello,
using an amplifier stage in front of the AD7799 would in principle allow you to use the full scale of the ADC. But you have to be aware, that this only increases the nominal resolution (refered to the weigh scale output), but it may decrease the effective resolution. The nominal resolution of the AD7799 is 24 bit. If you use only 6mV of the 39mV input range you loose ~3 bits of nominal resolution, but the remaining 21 bits are still more than the effective resolution you can achieve with the highest gain - the lost 3 bits would only display noise anyhow.
The effective resolution is limited by the noise of your setup. So if you want to improve your setup, you have to find an amplification setup with better noise performance than the instrumentation amplifier in the AD7799. This might be possible with an external instrumentation amplifier with very low 1/f noise (since in your application you look only at low frequencies), but it won't be trivial.
So you might in principle amplify your input signal with an "zero drift" instrumentation amplifier with significant better noise performance than the AD7799 amplifier. With the high external amplification you might reduce the gain of the AD7799 to get an improved effective resolution (table 7 of the AD7799 datasheet). But only a real noise analysis will tell you, if improved the effective resolution or if you deteriorated it.
Looking at the nominal resolution alone (i.e. use an external amplifier just to cover the fulls scale input without consideration of noise) will not improve your results.