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In this work, we extend the bottom-up reconstruction framework of $F(R)$gravity to other modified gravities, and in particular for $f(\phi)R$ andmimetic $F(R)$ gravities. We investigate which are the important conditions inorder for the method to work, and we study several viable cosmologicalevolutions, focusing on the inflationary era. Particularly, for the $f(\phi)R$theory case, we specify the functional form of the Hubble rate and of thescalar-to-tensor ratio as a function of the $e$-foldings number andaccordingly, the rest of the physical quantities and also the slow-roll and thecorresponding observational indices can be calculated. The same method isapplied in the mimetic $F(R)$ gravity case, and in both cases we thoroughlyanalyze the resulting free parameter space, in order to show that the viabilityof the models presented is guaranteed and secondly that there is a wide rangeof values of the free parameters for which the viability of the models occurs.In addition, the reconstruction method is also studied in the context ofmimetic $F(R)=R$ gravity. As we demonstrate, the resulting theory is viable,and also in this case, only the scalar-to-tensor ratio needs to be specified,since the rest follow from this condition. Finally, we discuss in brief how thereconstruction method could function for other modified gravities.

The Reconstruction of $f(\phi)R$ and Mimetic Gravity from Viable Slow-roll Inflation