Improving a badly conditioned matrix

Improving a badly conditioned matrix

I have a badly conditioned matrix, whose rcond() is close to zero, and
therefore, the inverse of that matrix does not come out to be correct. I
have tried using pinv() but that does not solve the problem. This is how I
am taking the inverse:
X = (A)\(b);
I looked up for a solution to this problem and found this link (last
solution) for improving the matrix. The solution there suggests to use
this:
A_new = A_old + c*eye(size(A_old));
Where c > 0. So far employing this technique works in making the matrix A
better conditioned and the resultant solution looks better. However, I
investigated using different values of c and the resultant solution
depends on the value of chosen c.
Is there an automatic way of selecting value of c for which I will get the
best solution?