We present an extensive new time series of spectroscopic data of the peculiar SN 1999aa in NGC 2595. Our data set includes 25 optical spectra between -11 and +58 days with respect to B-band maximum light, providing an unusually complete time history. The early spectra resemble those of an SN 1991T-like object but with a relatively strong Ca H and K absorption feature. The first clear sign of Si II lambda6355, characteristic of Type Ia supernovae, is found at day -7, and its velocity remains constant up to at least the first month after B-band maximum light. The transition to normal-looking spectra is found to occur earlier than in SN 1991T, suggesting SN 1999aa as a possible link between SN 1991T-like and Branch-normal supernovae. Comparing the observations with synthetic spectra, doubly ionized Fe, Si, and Ni are identified at early epochs. These are characteristic of SN 1991T-like objects. Furthermore, in the day -11 spectrum, evidence is found for an absorption feature that could be identified as high velocity C II lambda6580 or H alpha. At the same epoch C III lambda4648.8 at photospheric velocity is probably responsible for the absorption feature at 4500 8. High-velocity Ca is found around maximum light together with Si II and Fe II confined in a narrow velocity window. Implied constraints on supernovae progenitor systems and explosion hydrodynamic models are briefly discussed.