We present the full anatomy of various D-term extensions of the MSSM's Higgs sector, which lift the Higgs mass at tree level and for a range of tan beta. We then explore the LHC & ILC's capability to determine their effect through their enhancement of Higgs branching ratios with respect to the Standard Model. A given D-term enhancement of the Higgs sector allows one to further set a lower bound on the required stop masses versus trilinear soft term values to obtain the observed Higgs mass. In particular these extensions may bring stops within reach of the LHC, but with excesses in Higgs decays only just observable at the LHC or definitely observable at the ILC. We also comment on the D-term enhancement of MSSM squarks and sleptons.