The clip pictured above are large version which are designed to be able to stretch out to fit thicker table top widths.
These cast iron bird tablecloth weights are described as 'heavy weight' and are some of the heavier weights I have been able to find online. They're also quite attractive and don't look tacky like lots of other versions.
Or you could check out these clamps, which are spring loaded to fit firmly.
If you'd rather get a little bit more forceful (and hands on) then these D.I.Y tablecloth anchors may be the way to go.
D.I.Y Anchor with bungee cord and grommets
It takes a little effort, but is definitely a solution to the problem.
What you will need:
-Preferably three 12" bungee cords.
-Grommet kit (including cutting block, hole cutter and insert punch)
The edges of the tablecloth need to be folded over around 2 inches and bang the holes with the hole cutter using your hammer. Preferably, you'll place three along each side, approximately in the middle of your folded edge. After attaching the grommets according to the instructions provided with them, you simply have to stretch the bungee cord underneath to the grommet hoe on the other side.
Heavy weight tablecloths
You also might be interested in heavy tablecloths that are less likely to slip and slide around. Have a look around for something heavy enough to stay put.