An outer-independent double Roman dominating function (OIDRDF) of a graphGis a functionh:V(G)→{0,1,2,3}such that i) every vertexvwithf(v)=0is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertexvwithf(v)=1is adjacent to at least one vertex with label greater than 1, and iii) all vertices labeled by 0 are an independent set. The weight of an OIDRDF is the sum of its function values over all vertices. The outer-independent double Roman domination numberγoidR(G) is the minimum weight of an OIDRDF onG. It has been shown that for any treeTof ordern≥ 3,γoidR(T) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show that, for any connected graphGof ordernwith minimum degree at least two in which the set of vertices with degree at least three is independent,γoidR(T) ≤ 4n/3.