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)

 A: *

*The probability will not hit 0% or 100% exactly, it approaches them asymptotically.

*The uncertainty increases towards the extremes, which you will see if you include confidence intervals in your plot.

For these reasons, this is probably less of a problem than you think. But to answer your question: yes, there are alternatives to logistic regression. Probit regression is one, though it may actually be worse given your concerns, because the curve will approach the extremes more rapidly than with the logistic. GAMs can be used for binary data too. And even simpler, you can just use a moving-window mean.
A: 
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...
