Side Lobes in the Magnitude Response of a Low Pass FIR Filter
The filter is optimized to suppress signal components in the frequency ranges where the side lobes are low. The high side lobes in are just a consequence of optimizing the stopband suppression in those other frequency bands, where aliasing components are expected. The regions with high side lobes are basically "don't care" regions. If you wanted to increase stopband attenuation there, the stopband attenuation in the other - more important - regions would be decreased.