 Uniform ample subtraction from a big divisor with ample exceptional restrictions

If a big class $D=B+E$ has ample restriction to every component $E_i$ of the effective support, then for any ample $A$ a sufficiently small $ϵ>0$ makes $D-ϵA$ nef. Use openness of the ample cone to keep $B-ϵA$ and all $(D-ϵA){|}_{E_i}$ ample with one finite minimum of bounds. Curves inside the support use these restrictions; curves outside use effectivity of $E$. Thus one does not need a uniform choice over infinitely many subvarieties.