How to efficiently control an FIR's magnitude response by altering its phase spectrum
There are a few things that can make this easier


If you don't care about the phase response you can typically get the best result with calculating a minimum phase for your amplitude spectrum and using that as the complex target
If you want to control the in-between and the behavior at the band edges you need to specify targets points there as well. 
If you have more target points then FIR coefficients, you have an over determined system but you can still solve it with a least square error approach. this will not hit all your target points exactly but it's often a good compromise between filter fidelity and computational complexity. 
A least square error approach has the added advantage that you can apply weights to steer the solution towards a trade off that fits your requirements. For example, you typically need to control the out of  band behavior: you don't care about the exact behavior other than it shouldn't have too much gain. To achieve this, you can set the target simply to zero and play around with the weight until the gain in that band feels acceptable