An important consideration when designing a magnetostatic cavity for various applications is to maximize the ratio of the volume of field homogeneity to the overall size of the cavity. We report a design of a magnetically shielded solenoid that significantly improves the transverse field gradient averaged over a volume of 1000 cm3 by placing compensation coils around the holes in the mu-metal end caps rather than the conventional design in which the compensation coils are placed on the main solenoid. Our application is polarized 3He-based neutron spin filters, and our goal was to minimize the volume-averaged transverse field gradient, thereby the gradient induced relaxation time, over a 3He cell. For solenoids with end cap holes of different sizes, additional improvements in the field gradient were accomplished by introducing non-identical compensation coils centered around the non-identical holes in the end caps. The improved designs have yielded an overall factor of 7 decrease in the gradient in the solenoid, hence a factor of 50 increase in the gradient induced relaxation time of the 3He polarization. The results from both simulation and experiments for the development of several such solenoids are presented. Whereas our focus is on the development of magnetically shielded solenoids for 3He neutron spin filters, the approach can be applied for other applications demanding a high level of field homogeneity over a large volume.