There is a case to be made either way.
This is a great question, and it is going to come down to a DM ruling, as there is a compelling case to be made for either ruling. Unfortunately, I cannot give a definitive answer either way, but I can offer some arguments and let you decide. The first is a more strict rule-oriented approach, that is, let’s just go by what is written without trying to make sense of it; and the other is a more “simulationist” approach, that is, what makes the most sense in the context of the narrative. The DM and the players should just work out how they want to rule on the Robe, and apply that ruling consistently over the course of the campaign.
Interpretation 1: You can still see in all directions, even while the robe is covered by another article of clothing.
There is a case to be made here based on the Robe’s interaction with creatures that have abilities that trigger when they are seen. The medusa has an ability called Petrifying Gaze:
When a creature that can see the medusa’s eyes starts its turn within 30 feet of the medusa, the medusa can force it to make a DC 14 Constitution saving throw if the medusa isn’t incapacitated and can see the creature.
To avoid this, a creature can usually avert its eyes:
Unless surprised, a creature can avert its eyes to avoid the saving throw at the start of its turn. If the creature does so, it can’t see the medusa until the start of its next turn, when it can avert its eyes again.
While wearing the Robe of Eyes, a creature is never considered to be averting their eyes:
Although you can close or avert your own eyes, you are never considered to be doing so while wearing this robe.
If you can never avert your eyes from the Medusa, you can always see the medusa, even while wearing another article of clothing over the robe.
Interpretation 2: The eyes on the robe are doing the seeing, so they would see only the article of clothing that covers them.
Alternatively, we can make an argument from the spell description that the eyes are function as points of sight, and what you see is relative to their position on the robe. Usually magic items don’t tell us how they work – magic be magic. But with the Robe, it seems to indicate how it works:
The eyes on the robe can’t be closed or averted.
This seems to indicate that the position of the eyes matters, so covering the eyes with another robe would mean they see the inside of the robe only.