The effect of doping of thin organic glassy films of the hole transport material $N$,${N}^{\ensuremath{'}}$-di(1-naphthyl)$N$,${N}^{\ensuremath{'}}$-diphenylbenzidine ($\ensuremath{\alpha}$-NPD) with various concentrations of 4,${4}^{\ensuremath{'}}$, ${4}^{\ensuremath{''}}$-tris($N$-(1-naphthyl)-$N$-phenylamino)-triphenylamine (1-NaphDATA) is investigated with respect to the resulting charge carrier mobility and the effect on thermally stimulated currents (TSC). The energetic trap structure is obtained utilizing the fractional TSC technique. At low doping concentrations of a few percent the activation energy of an about 0.5 eV deep trap level is measured, which is related to the higher lying highest occupied molecular orbital level of 1-NaphDATA compared to the $\ensuremath{\alpha}$-NPD matrix. A strong shift of the TSC peak towards higher temperatures with increasing dopant concentration is observed. It can be attributed to re-trapping despite the low film thickness of 200 nm. Thereby the effect of re-trapping is theoretically investigated and found to be composed of a mobility change and a change of the trap depth, as it is measured in fractional TSC measurements. The latter is related to enhanced re-trapping into deeper traps. The maximal peak shift occurs at about 4 vol % of dopant concentration, corresponding to about 2.7% molecular concentration. For higher concentrations of 1-NaphDATA the mobility increases again, and the TSC peak gradually changes to that of pure 1-NaphDATA, explainable by a progressive change from a deep-trap-controlled transport to a percolating transport on the 1-NaphDATA molecules.