In addition to my answer to What's the smallest change to physics to allow magic, which suggests there may be more magic in today's world than we oft believe, there are some approaches you can take to to make your ubiquitous magic seem more magical and less scientific/technological.
One of the key aspects of science is the search for the so called "natural laws." These are aspects of reality which are true everywhere and at every time. Science tries to identify these laws, if they truly exist, so that we can act on them. However, there are known patterns which science is particularly poor at predicting. One of these classes of patterns is those which include the actions of the observer. In science, it is utterly forbidden for the observer of a scientific experiment to influence its effects once the experiment is underway. They are expected to set up the initial state and then let the experiment simply evolve according to the "natural laws." This is then used to distill the essence of those natural laws into equations we can use. If we permit the scientist running the experiment to interact with it, we run into all sorts of issues like confirmation bias that the fundamental scientific process simply cannot deal with. Scientists take great effort to make sure their science is not plagued by this with techniques such as double-blind testing and repeatable experiments.
So what if your magic didn't follow this rule? What if it was simply impossible to use magic without fundamentally altering the rules of magic in the first place? The alteration might very slight, but perhaps the cumulative sum of all of the magic use in your world causes the laws of magic to slowly change shape. If they did so, science would never find a "natural law" which never changes, so they would have to reject the hypothesis that such a law exists. However, if the changes were smooth, you would find that magic didn't change 'all that much' over the short run This would be enough for everyone in your society to use a particular magic, but find that it eventually fails them.
It would also open an interesting door for non-scientific approaches if magic changed differently in different areas. Perhaps, in one kingdom, there's one particular kind of magic which changes slowly within its extent. Outside of the kingdom, that kind of magic changes much more rapidly. This would create a unique flavor of magic for that kingdom because they would have more time to codify "scientifically" this magic's behavior. Another kingdom may have a completely different feeling.
I envision this having a feeling similar to surfing. We all know the ocean's surface is flat, but every now and then a wave comes by. As long as you ride where the wave wants you to go, you're surfing. And, from what I understand, its a feeling quite unlike anything else. You would have groups of individuals chasing magic waves around the world, riding them as they go.
However, some individuals would see deeper than that. While most will happily ride along on the waves created by millions of magic users around the world, others may realize that their actions have side effects -- especially around some key nodes where magic is unusually stationary. These individuals would plumb the part of magic which science could not touch: the kind which can only be had by entering that magic and experiencing it for yourself -- manipulating it and having it manipulate you. Magic may behave quasi-scientifically elsewhere, but near these nodes, the mages would rule the day, not the scientists.