Is it possible and good to know and/or talk about other/more details about the way out of suffering than the Buddha did? (One of the reasons for asking this question is that I thought that maybe some people needed some extra/other knowledge/insight in order for the Noble Eightfold Path to become more accessible to them.)

Not necessarily to be "better" in knowledge and understanding than the Buddha overall, but to have a better understanding or additional explicit knowledge of some things or see certain things in more detail or accuracy with regards to certain aspects about the way out of suffering, and that may also address particular, "extreme" types of ignorance, mental illness, psychosis.   

To be able to answer other questions related to the way out of suffering that the Buddha was never asked.
In the parable of the Simsapa leaves in SN 56.31, the Buddha took some Simsapa (Indian rosewood) leaves in his hand and said that the leaves in his hand were far less in number than the leaves in the forest. Similarly, he had realized far more than he taught. However, that which he taught is all that is needed for the path to end suffering. That which he did not teach is not relevant to the path to end suffering.

Meanwhile, in DN 16, he declared that he did not hide (in a "closed fist") any teachings from his disciples. He said he did not separate some teachings as being exotic or esoteric.

Based on these statements, we can say that more details regarding the path is unnecessary. Also, since the Shakyamuni Buddha was the last Sammasambuddha (or Buddha with full perfection), it is unlikely that anyone else coming after him would know more of the path than he does, unless they too achieve the same level of perfection.

Hence, it is plausible that primarily relying on the Buddha's original words (Buddhavacana or the Buddha's Words) in most of the Sutta Pitaka (and the corresponding Mahayana Agamas) is sufficient for understanding his teachings and the path. I say "most", because there are parts of the Sutta Pitaka which contain teachings or writings from other people e.g. Sariputta, Ananda, Mogallana, elder monks, elder nuns, a few advanced householders etc. Some parts of the Sutta Pitaka like some books in the Khuddaka Nikaya even come from later periods, like the Milindapanha. You can find more scholarly information on the authenticity of the Buddha's original words in the book "The Authenticity of the Early Buddhist Texts" by Bhikkhu Sujato and Bhikkhu Brahmali.

But this does not mean that it's wrong to elaborate or expand on the Buddha's teachings, while not contradicting the original teachings. In fact, I think teachers over the ages would definitely try to explain the teachings using new methods and analogies. However, such elaborations or expansions should be considered secondary sources, which are less important compared to the Buddha's original teachings.

Many traditional Theravadins say that the Abhidhamma was taught by the Buddha, but academic scholars say that this is a later work by disciples. To me, I would consider the Abhidhamma as a secondary source, that explains the Buddha's original teachings in the Sutta Pitaka. The many carefully collated lists in the Abhidhamma are useful.

The Visuddhimagga (or Path of Purification) written by Buddhaghosa in the 5th century in Sri Lanka is another useful secondary source that expands on Buddhist meditation practice and doctrine. It can be downloaded here.

Also many traditional Mahayanists say that the Mahayana sutras were taught by the Buddha in the second and third turning of the wheel, but academic scholars say that these are later works by Mahayana authors. To me, I would consider these as secondary sources too. Nagarjuna's Mūlamadhyamakakārikā contains an expansion on the Buddha's original definition of emptiness. I tried to analyze the link between Madhyamaka emptiness and Theravada emptiness in this question.

Of course, I understand that there will be traditional Theravadins and traditional Mahayanists who will disagree with some of my opinions above.