One way to beat the problem (still fairly radical) is to go active. With an active preamp system in your guitar and a set of pickups such as Duncan LiveWires, you cure the problem.
But there is another solution short of trading in your guitar or routing the bodyy out to make room for humbuckers or active electronics. There is a group of pickups designed to be hum-canceling or "bucking," that retain the essential single-coil sound. Seymour Duncan makes several. Duncan’s Classic Stack pickups (neck, middle) have the dual coils of a humbucker, but they are stacked so that the pickup has the dimensions of a single-coil meaning no mods are required to your guitar. They also retain the sound of a single-coil, but without the hum.
The Seymour Duncan Vintage Rail pickup is another that accomplishes the same mission offering single-coil sound without the hum. Yet another pickup that is growing in popularity is the Virtual Vintage from DiMarzio.
Another aftermarket option is the Lace Sensor Gold 3-Pack that puts out vintage tone with a glassy, bell-like sound and crisp highs. Lace also offers their Gold pickups prewired in a Strat pickguard and a 3-pack with a revised version of the Gold pickup with a hotter bridge pup and stronger bottom and singing midrange.
Even closer in tone to your vintage single-coils are Fender's new Vintage Noiseless pickups. Fender's noiseless technology has very successfully minimized the noise while retaining the tone. They’re offered in a 3-pickup set for Strats and in a 2-pickup set for Telecasters.
This isn't an exhaustive list, but long enough to demonstrate there are alternative ways to keep your single-coil-equipped guitar "as is"; producing vintage-like tone, but with sweet silence rather than buzz between the notes.