While there are a few ways to add an “invisible” signature, some of these ways might not work as intended with different PDF tools.
The best, most compliant, way to add a signature that is not visible is to add an Author/Certification signature, which can be thought of as a document wide signature added by the original document author. These can have be given a widget on page 1 with zero width and height, and no appearance. Other PDF readers should treat this as you expect.
Note, there can only be one author/certification signature, and it must be the first signature.
See the CertifyPDF function in the following sample, and just change the following line of code
SignatureWidget widgetAnnot = SignatureWidget.Create(doc, new Rect(143, 287, 219, 306), certification_sig_field);
to
SignatureWidget widgetAnnot = SignatureWidget.Create(doc, new Rect(0, 0, 0, 0), certification_sig_field);
and comment out this part
// (OPTIONAL) Add an appearance to the signature field.
Image img = Image.Create(doc, in_appearance_image_path);
widgetAnnot.CreateSignatureAppearance(img);