Nope don't check for it...but that is just because it is so obvious it is there that no need to. If it were not there then I would suddenly pay attention, like super fast! However, this is purely a function of huge amounts of time with the same EEEeeeeeeeeeeeeeeeee tone, for decades. So indeed it eventually 'becomes the equivalent of my silence'. In short, that's what "silence" is to me...a screaming E-flat.
If it goes up in volume then the "checking" starts again. It's normal to do that at first I think, so I wouldn't get too flustered with oneself. Time is a very personal thing and it can be 'longer' for some than others.
Note though, that my T is very stable and tons of you have variable T, which is different...that would obviously be more prone to "where are we today in T world???!!!".
My T 'stability' is destroyed by reactivity to sound, and H...But they don't happen when I wake up as my room is quiet and my house is quiet. It's life out there that is not quiet!
But "waking up"...Nope, that's not a T issue now.
Best, Zimichael