Regression coefficient interpretation in binary logistic regression

I have performed a binary logistic regression with re-contracted as the DV (not re-contracted =0, re-contracted =1).

Team ladder position at the end of the year is a significant predictor. This varible is a rank (1-18) with 1 being "top" of the ladder and "18" being the bottom. The regression coefficient for this variable is -0.044 and the exponentiated coefficient is 0.957.

I am confused by the interpretation given that that a one unit increase in the predictor variable would actually be a worse result e.g. a higher number is toward the bottom of the ladder. Does this then mean that "a one unit increase in ladder position (towards the bottom) reduces the probability of being re-contracted by -0.044? and thus, the odds of being re-contracted are reduced by 0.957 for each one unit increase in ladder position (towards the bottom of the ladder)?

I'm going round in circles! have I interpreted correctly?

No. The interpretation would be that a one-unit increase in ladder rank is associated with an expected (1 -.957)% = 4.3% decrease in the odds of being contracted. To obtain the probability at rank level $X$, you'd need to calculate, $\hat{p}$=$exp(X\hat{\beta})\over{1+exp(X\hat{\beta})}$.