Galileo is the first global navigation satellite system to authenticate their civilian signals through the Open Service Galileo Message Authentication (OSNMA) protocol. However, OSNMA delays the time to obtain a first position and time fix, the Time To First Authentication Fix (TTFAF). Reducing the TTFAF as much as possible is crucial to integrate the technology seamlessly into the current products. In the cases where the receiver already has cryptographic data available, the so-called hot start mode and focus of this article, the currently available implementations achieve an average TTFAF of around 100 seconds in ideal environments. In this work, we explore the TTFAF optimizations available to general OSNMA capable receivers and to receivers with a tighter time synchronization than the required by the OSNMA guidelines. We dissect the TTFAF process, describe the optimizations, and benchmark them in three distinct scenarios with recorded real data (open-sky, soft urban, and hard urban) and the official OSNMA test vectors. The first block of optimizations centers on extracting as much information as possible from broken sub-frames by processing them at page level and combining redundant data from multiple satellites. The second block of optimizations aims to reconstruct missed navigation data by the intelligent use of fields in the authentication tags belonging to the same sub-frame as the authentication key. Combining both optimization ideas improves the TTFAF substantially for all considered scenarios. We obtain an average TTFAF of 60.9 and 68.8 seconds for the test vectors and the open-sky scenario, respectively, with a lowest TTFAF of 44.0 seconds in both. Likewise, the urban scenarios see a drastic reduction of the average TTFAF between the non-optimized and optimized cases. These optimizations have been made available as part of the open-source OSNMAlib library on GitHub.