  We further develop and extend a recent perturbative framework for neutrino
oscillations in uniform matter density so that the resulting oscillation
probabilities are accurate for the complete matter potential versus baseline
divided by neutrino energy plane. This extension also gives the exact
oscillation probabilities in vacuum for all values of baseline divided by
neutrino energy. The expansion parameter used is related to the ratio of the
solar to the atmospheric $\Delta m^2$ scales but with a unique choice of the
atmospheric $\Delta m^2$ such that certain first-order effects are taken into
account in the zeroth-order Hamiltonian. Using a mixing matrix formulation,
this framework has the exceptional feature that the neutrino oscillation
probability in matter has the same structure as in vacuum, to all orders in the
expansion parameter. It also contains all orders in the matter potential and
$\sin\theta_{13}$. It facilitates immediate physical interpretation of the
analytic results, and makes the expressions for the neutrino oscillation
probabilities extremely compact and very accurate even at zeroth order in our
perturbative expansion. The first and second order results are also given which
improve the precision by approximately two or more orders of magnitude per
perturbative order.
