A few character cards talk about the day of the week. That is correct. These characters are useful in the beginning of the week, but as the week goes on they get increasingly more costly to visit again.
I assume that is essentially the round you are in?
So for example, Adler comes out on the Day 4 spot, but you are now in day 6 (round 6), it costs 6 investigation markers to use her (as opposed to 4 - the space her card is on). Correct?