My website has currently 3 CSS files that are automatically included as a part of the website and I do not have access to the source i.e. index.html
In your new CSS file, add !important to the block of code, for instance:
if you have this on css1.css:
h2{
color:#000;
}
In css4.css put the same element, but with !important, as follows;
h2{
color:#ccc !important;
}
Therefor, !important will force that this style will be by all means the style set for the element to which it's applied.