Are there truths that the scientific method cannot explore, but maybe alternative methods can? Does the scientific method subsume all possible valid ways to obtain true knowledge? Or are there alternative methods to explore truths that lie beyond the space of truths reachable by the scientific method due to its methodological constraints?
One avenue to consider might be phenomenology. Phenomenology is the study of the nature of experience from the first-person point of view. It was first articulated by Husserl and was developed by Heidegger and Maurice Merleau-Ponty.
What distinguishes phenomenology from the objective or natural sciences is the emphasis on the first-person experience, whereas scientific methods generally deal with objective analyses.
Phenomenology has generally received more attention in Contintental (i.e. European) rather than Anglo-American philosophy. One influential book to look up is The Embodied Mind: Cognitive Science and the Human Conditon. It combines insights from phenomenology with cognitive science and elements from Buddhist abhidharma (philosophical psychology).
There's also an interesting current essay on Merleau Ponty on Aeon https://aeon.co/essays/the-phenomenology-of-merleau-ponty-and-embodiment-in-the-world. (His major work, Phenomenology of Perception, is an exceedingly difficult read, in my experience.)
Finally there's a good general introduction,Routledge Introduction to Phenomenology, ed. Dermot Moran, which has good summaries of many of the major thinkers and ideas.
Definitions if science are similar to this:
the intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment.
So these are truths about the physical and natural world today. One limitation is that we cannot observed things that happened in the past (unless in cosmology), but we can learn supposed truths by reading witness accounts. This is common also in law.
Mathematical and logical truths are found by applying logic and formalism to prove statements to be correct.
Another problem is that experimentation may not be possible for to legal, ethical, economical or practical concerns. For some topics, modeling and simulation may lead to truths. For psychological topics, certain forks of therapy, hypnosis or medication might reveal some truths hidden in the mind.
Qualia are also difficult for science. As an example whether music or a painting express sadness, would typically be investigated indirectly by asking instead how many people would agree to such emotional expression.
So called supernatural or paranormal phenomena may not be investigable, and diverse claims for achieving truths have been made, such as prophets hearing truths as choices in their heads, but it seems impossible to distinguish truths from inventions in that area (faith means not requiring evidence).
The question seems a bit broad.
Yes, of course "truths" and orthodoxies and certainties can be explored by "other methods" and have been for millennia.
They can be explored by dialectic in the Socratic manner, by inward investigation in the Zen manner, by interpretation in the Hermeneutical manner, or by external examination in the manner of the Catholic Inquisition. Indeed, nearly all metaphysics, logic, ethics, and theology can be said to investigate "truths" by methods other than those dubbed "scientific."
Nothing is preventing you from taking up these methods. But they probably will not settle many disputes or appear in scientific journals. They will not satisfy the highly influential protocols for settling disagreement found in modern science.
Science simply works with demonstrable effects and refers disputed hypotheses to ongoing demonstration, evidence of the senses. What works is what counts. Newton demonstrated a constant effect he called "gravity." He famously declined to further explain what it is or why it is.
I believe you are attacking a straw man called "Science as Absolute Truth." You can make compelling arguments for "spiritual" events or "miracles" to your heart's content. Many authority figures do and are widely believed. But by the very definition of such claims you can't repeatedly demonstrate them under the bright lights of the lab, so to speak.