There's no special secret way to doing it; you gotta get a wire from the bridge to [wherever you're sticking your ground wires, usually the back of a pot]. There's no replacement for a bridge ground wire, no other part you can run a wire to. There has to be a wire under the bridge, no other way.
You can either do it the quick and very messy way of dragging a wire up through the bridge pickup cavity and just running it along the top of the body to the bridge, which usually causes the pickguard to not sit flat and generally looks terrible, or you do it the proper way and drill a hole through to the control cavity.
What I do on my (hardtail) bodies is I use a slightly larger dill bit, usually around 6mm or so, to drill out just a vertical hole right under the bridge, about 5mm deep, then use a smaller bit like 3mm to drill a diagonal (as horizontal as I can get it) hole from that first larger hole through to the control cavity. That way I know there's a good few millimetres of wood above the wire and I'm not at risk of drilling through the top, plus it means there's a little extra room under the bridge to accommodate my sloppy big blobs of solder.
With a Strat, specifically, you can also drill from just inside the front edge of the bridge into the pickup cavity, which is usually a shorter distance to drill than going from the middle of the bridge to the control cavity. Plus if you screw it up there's usually very little of the guitar showing on that part of a Strat, so between the bridge and the pickguard any accidental scratches or haphazard drilling will be hidden.
