The problem of holding thin work for operations on the whole upper surface is an old one and various methods have been developed to keep the job in place. Unfortunately naming conventions are, um, “inconsistent”.
Time to research toe clamps, edge clamps and hold-downs for something either affordable or DIY friendly. Unfortunately searches for hold-downs generate a lot more wood worker specific information than metal worker stuff.
There are various devices using a screw and wedge principle to generate both sideways and downwards forces. For example Mitee-Bite sell most versions. Variations on the eccentric tightened style appear to be the most favoured DIY version.
Its often necessary to put the job on a flat spacer to make room for the clamping device to work.
The primary old style devices were hold-downs used in conjunction with a vice or solid stops to generate the necessary forces. Basically a shallow triangle with the shallow end having either a sharp edge or small radius and the deep end either angled a bit over 90° relative to the base (92° – 95° are typically specified) or having a larger radius. Pushing the device down against the stop with the sharp end a bit above the middle of the job thickness generates both downwards and horizontal forces needed to keep things in place.
Couple of pictures found on the internet of the Starrett 54c hold downs. Commercial product and slightly more sophisticated than a simple wedge. The size pictured is about 4″ long
End profile

In use

One oft confusing point about hold downs is that they can be made to work either pointing downwards, which intuitively appears more correct, or pointing upwards as in the picture above. The angle on the short side defines which way up they work. Putting them in upside down will lift the work. All the textbook diagrams I’ve seen show the pointing down variety but Mr Starrett must have had a reason for going the other way.
Clive