I'm going to guess that the major factor is breathing. High notes in particular really need good breath support, and your workout is prepping you to take full breaths and have good posture for breathing. It also sounds like you're coming away from your exercise with an "energized" feeling. Singing is a physical process, but it involves a coordination of things. Mental state can have a big effect on that.
Is there another way? Yes. If there are notes that you can hit with ease after you exercised, you should be able to hit them just fine even if you haven't exercised. The fact that you're not suggests a possible issue with your normal technique. Do you feel like you're straining to reach the higher notes normally? Do you feel like you're "reaching up" to them, or struggling to get them out? If you are, that can be fixed.
I would recommend:
Take in really deep breaths for singing, especially high notes. Practice singing them after breathing in as deeply as your can. Take your time if you need to -- in performance you'll need to take in deep breaths fast and sometimes for lengthy lines of vocals. But for the purposes of practice, focus on supporting the tone -- you want to feel like you have more than enough air for any note you sing.
Make sure your larynx is relaxed. When "reaching" for a high note, there is a tendency to raise your larynx, which leads to less space in your throat and more strain. Your larynx will rise a little bit as you sing higher, but you want to minimize this. From your description of how you feel after you exercise, I am confident that your larynx is pretty relaxed at that point... you want to make sure that's the case before exercise, too.
Try visualizing the notes differently. If you feel you are reaching for them, try imagining that you're floating on top of them. Or that you're grabbing them with a solid hold. Or that you're hitting them right on the bullseye. Or whatever you come up with on your own. It might sound kind of silly, but visualizing can help.
You can also try imagining that you have just exercised. Imagine the feeling of power and energy you have. Or, next time you're singing after a workout, pay attention to what you're thinking in your mind as you conquer those high notes. Then try to replicate that when you practice.
Those things should help with technique. The other major thing to think about is your vocal warmup. You mention doing hisses, lip trills, humming, and scales, and all of those things are good, but the first three things are probably not specifically very helpful for high notes (or capturing that seize-the-day feeling). I think scales are the most relevant. When you're first warming up I'd recommend starting conservative, but then expanding to really go across your entire range. (For example, my range is about D3-D6, but I start with a round of exercises going from about A3 up to F5 or so, then I come back down to F3, then go back up to A5, then down to D3, then up to C6. If I go up too high too early, I strain myself a bit and lose the low notes.)
Another thing to think about is length of warm up. Experiment with warming up for different amounts of time... and with different exercises, across different ranges. You might decide to have some practice sessions where you only sing warmups the whole time. Or, you might just do a few exercises then go right into songs. Songs are less boring than exercises, but also not as good for things like working your way up to the high notes. But you might discover that certain songs are particularly helpful to you for practicing certain things. (For example, I have a friend who tended to struggle to not go flat with a B4 in one song, but he could soar up to an E5 no sweat in another. Clearly, it wasn't the fact that the B4 was "high" that was the problem, even though he had the subjective feeling of it being high. The B4 in this case was much, much longer note that required a much bigger breath at the beginning and more skill in sustaining the note evenly. The E5 was good practice for his higher range. Both songs were good practice in their respective ways.)
That's about all I can think of for the moment, but hopefully it's given you some things to think about. If you can identify what you're doing differently when singing before and after a workout, you can certainly get to that point much faster and with less sweat.
Edit: One more thing. If you don't currently have a vocal coach or instructor, I'd definitely recommend one. The immediate feedback you can get on your singing can be invaluable!