ALBANY—Support is growing in the state Legislature for a full-scale legalization of medical marijuana, an issue that likely will be among a handful of battles in the Capitol over the next few months.

In January, Gov. Andrew Cuomo introduced his own more limited effort to make pot available to the very ill. But in the months since then, more than a half-dozen state senators who had previously publicly opposed full-scale legalization have announced their support for a bill to allow health-care practitioners to prescribe the...