Would there be any alternatives to a Logistic Regression or way to modify the Regression for what I'm looking for? To provide more information I am looking for an alternative to logistic Regression or a way to modify it. This is because of two reasons:

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*My data is widely dispersed across the X axis for both my 1s and 0s however there is slightly more 1s the higher the X and slightly more 0s the lower the X. So the probabilities listed in a standard Logistic regression I feel are over/understated on both ends of the X axis.

*I'm not very concerned with classifying future outcomes accurately, I'm more concerned in how movement along the X increases the probability for my 1 outcomes.

My main concern with a Logistic Regression is that the starting off point the probability of a 1 is close to 0% while the ending point the probability is 100% (given a positive relationship with the x).
With my data this isn't accurate as the probability of a 1 outcome would never hit 100%. I am wondering if there would a better method for me to utilize that would allow me to map out the relationship between my X variable against my 1 and 0 Y variables? Or a way to modify the regression to take into account that for my max X variable I will never have a 100% probability of getting a 1?
Basically I'm interested in knowing what my probability of a 1 outcome is for my max X while getting an accurate representation of how the probability will change with movement across my X axis.
Edit: Picture for example of a Logistic Regression (Not a picture of my actual data though)



My data is widely dispersed across the X axis for both my 1s and 0s
however there is slightly more 1s the higher the X and slightly more
0s the lower the X. So the probabilities listed in a standard Logistic
regression I feel are over/understated on both ends of the X axis.

I feel you have misunderstood logistic regression.  The model is fit to your data, so if at extremes of your training data you don't approach 0/100% in the raw data, then you won't with logistic regression either. But clearly if your training data is between +/-1 there will be some range outside your data set where you approach 0/100%, maybe its at +/-10, maybe its at +/-100...
