I'm not sure I know what the goals are here, but I'll go.
Instead of increasing global contrast, try a piecewise approach.
% cropped from a low-quality JPG screenshot
inpict = imread('IMG_2294.jpeg');
inpict = im2gray(inpict);
% curve parameters
hw = 0.1; % half-width of central portion (0 <= hw <= 0.5)
kc = 1.5; % slope in central region (kc >= 1)
kt = 2; % slope of tails (kt >= kc)
% construct the curve
pax = 0.5 + hw;
pay = 0.5 + kc*hw;
pbx = 0.5 - hw;
pby = 0.5 - kc*hw;
C = [0 0; (pbx - pby/kt) 0; pbx pby; pax pay; (pax + pby/kt) 1; 1 1];
% plot the curve
plot(C(:,1),C(:,2))
unitaxes()
% apply the curve to the image
outpict = im2double(inpict);
outpict = interp1(C(:,1),C(:,2),outpict,'linear','extrap');
outpict = min(max(outpict,0),1);
% compare the results
montage({inpict outpict})
With this example, the contrast near the central pivot can be controlled independently of the contrast in light and dark regions. In this case, I only applied a slight boost near gray, but a much stronger boost on the tails. Compare the tail positions of the contrast curve with the distribution of values on the input.
% look at the input distribution
imhist(im2double(inpict))
The result is minimal clipping, though that amount can be adjusted with the curve parameters.
