John Webster (‘One Who Is Son: Theological Reflections on the Exordium to the Epistle to the Hebrews’ in The Epistle to the Hebrews and Christian Theology, p 88) has some insightful things to say about how Hebrews 1.1-4 helps us locate the doctrine of providence:
It has its place in a larger sequence of teaching about, first, the immanent lively relations of Father and Son, and then, second, the Son’s agency as creator, saviour and Lord. Providence is shaped by theology proper, by Christology and soteriology. It is not an aspect of a theistic cosmology or a theory of history, but a corollary of the Father’s dealings with creatures in the Son.
Yet the force of this recognition ought not to shrink our understanding of the Son’s providential work (pp 88-89):
This, of course, does not collapse the properly cosmic scope of providence into the domestic sphere of Christian experience. Bearing all things, the Son’s governance is comprehensive. The one who bears all things is their maker. And his relation to all that he makes is not merely originary, as a kind of initial impulse; it is enduring and purposive. Having brought the creation to be, he enables it to continue to be, making it the object of his continual care.
How awesome that this is true about the One who made purification for our sins and is seated at God’s right hand!