Peaches are consumed worldwide and have great market demand. Compared to apricots, the volatiles defining the typical peach aroma are still poorly analytically characterized. The aim of our study was to determine the impact of the stage of ripening, sample storage conditions, and type of fiber coating and extraction glassware on peach aroma compounds. The peach aroma components were extracted by headspace solid-phase microextraction (HS-SPME) and with the application of a specific fiber (DVB/CAR/PDMS fiber) that retained the main group of volatiles present in peaches. Artisan-made glassware that enabled bigger sample loads was used. It was found that its application provided a 3-fold higher extraction of aromatic compounds on average. Significant differences were also found when the same peach pulp was analyzed fresh or frozen (in liquid nitrogen or in the freezer at -16 °C). In fresh peaches, a higher amount of some alcohols and important esters was determined. Liquid nitrogen had a positive impact on hexanal and (E)-2-hexenal, whereas the storage of samples in the freezer had a major impact on most other aldehydes, as well as limonene. The study revealed the importance of sample preparation and storage on the overall aromatic profile of peaches.