I'm reusing the alpha channel indeed. And I reverse it for the innermost side of the border.

    const float MIN_ALPHA_THRESHOLD = 0.3;
    const float MAX_ALPHA_THRESHOLD = 0.7;

    [...]

    if (fragmentColor.a < MIN_ALPHA_THRESHOLD) {
        // Outer edge of the border
        float outputAlpha = fragmentColor.a / MIN_ALPHA_THRESHOLD;
        outputColor = vColor * outputAlpha;
    } else if (fragmentColor.a > MAX_ALPHA_THRESHOLD) {
        // Inner edge of the border
        float outputAlpha = 1 - ((fragmentColor.a - MAX_ALPHA_THRESHOLD) / (1 - MAX_ALPHA_THRESHOLD));
        outputColor = vColor * outputAlpha;
    } else {
        // "Inside" of the border
        outputColor = vColor;
    }

So if the border goes like transparent->opaque, I divide it into segments using a threshold (transparent->min_threshold->max_threshold->opaque) and change the alpha values:

- Smoothen out the transparent->min_threshold segment, so that it goes linearly from a=0 to a=1

- make opaque (a=1) the min_threshold->max_threshold segment

- Revert and smoothen out the max_threshold->opaque segment so that it goes linearly from a=1 to a=0