The spell etherealness teleports you only to the border ethereal, and every source I've read about the deep ethereal says you get there via the border ethereal when going through a "curtain".
What does that mean?
The 5e DMG describes the Ethereal Plane in more detail in the Creating a Multiverse chapter. However, you seem to have the direction of travel the wrong way around. The Deep Ethereal cannot be reached from the Border Ethereal directly; travelling to the Deep Ethereal requires more powerful magic, and from the Deep Ethereal, one can travel back out to the Border Ethereal overlapping with other planes by passing through the aforementioned curtains.
The DMG says specifically:
Characters can use the etherealness spell to enter the Border Ethereal. The plane shift spell allows transport to the Border Ethereal or the Deep Ethereal [...]
To reach the Deep Ethereal, one needs a plane shift spell or arrive by means of a gate spell or magical portal. Visitors to the Deep Ethereal are engulfed by roiling mist. Scattered throughout the plane are curtains of vaporous color, and passing through a curtain leads a traveler to a region of the Border Ethereal connected to a specific Inner Plane, the Material Plane, the Feywild, or the Shadowfell. The color of the curtain indicates the plane whose Border Ethereal the curtain conceals; see the Ethereal Curtains table.
None of the text suggests it is possible to travel to the Deep Ethereal from the Border Ethereal, only the other way around. So, to get to the Deep Ethereal, you need to use the spell plane shift, the spell gate, or else some other kind of magical portal which leads there directly.
However, this has not been consistent throughout D&D's history. For instance, the 2e Planescape resource, A Guide to the Ethereal Plane, explains that travellers in the Border Ethereal can indeed pass through into the Deep Ethereal, simply by achieving the right mental state and willing it so:
A body’d [person] think that such a journey would be difficult, but it just ain’t so. Leaving behind the Border Ethereal is as simple as willing it, so canny cutters need to limit any wayward thoughts. [...]
A body on the Border who witnesses someone else leaving the Border for the Deep sees the traveler suddenly obscured in a roil of multicolored mists that finally boil away into nothingness, leaving no sign of the traveler.
From the Deep side, the traveller can see the wall of colour, a "vast, undulating curtain of vaporous color" (some sources describe the curtains as individual features in the Deep Ethereal, others as parts of a single, infinite, many-hued wall). This wall is described in further detail in the next chapter:
As mentioned in the Border Ethereal chapter, the curtain of vaporous color is a shimmering wall of ethereal mist that leads to specific areas (adjoining planes) on the Border. The infinite areas of the wall are the shoals and reefs of the Waveless Sea, and they languidly roll, ripple, and flap like world-sized banners - though a body can only see a small fraction of each area. [...]
The curtain of vaporous color is unique to the Deep Ethereal, and if a berk who thinks he's on the Border sees it, he's fooling himself about his location. Fact is, he's lost. Once he steps through a curtain, the traveler is on the Border Ethereal, but he can no longer see the curtain.
I would describe the wall, or a curtain, as an area where the roiling mists of the ethereal plane suddenly and sharply grow so thick as to prevent any sense beyond, but shimmering with specific colour as if lit from behind by some ubiquitous light source, reflected and refracted by the mist of the curtain; the area of curtain/wall itself slowly undulates, like fabric caught in wind, albeit on a grand scale.
To reach the Deep Ethereal, one needs a plane shift spell or arrive by means of a gate spell or magical portal.
The curtains are for travelling from the Deep Ethereal to the different Border Ethereal regions:
Scattered throughout the plane are curtains of vaporous color, and passing through a curtain leads a traveler to a region of the Border Ethereal connected to a specific Inner Plane, the Material Plane, the Feywild, or the Shadowfell. The color of the curtain indicates the plane whose Border Ethereal the curtain conceals; see the Ethereal Curtains table.