As long as the adjective matches its noun in case, number, and gender, is it possible to move the adjective anywhere in a sentence, even outside of prepositional phrases and subordinating or coordinating clauses?
That's a large question! Whole books were written on Latin prose composition in the days when it was taken seriously in teaching the subject.
In principle, yes, you can do as you suggest, but there are rules and, unless you are very careful, you might alter the meaning significantly. For example, Gaius hastam magnam fert simply means "G. is carrying a large spear", but hastam Gaius fert magnam is "G. is carrying a spear that is large", drawing more attention to the weapon's size; in Marcus Juliam amat, the verb is less emphatic than in amat Juliam Marcus. This also applies to prepositions, with magna cum laude meaning more than cum laude magna.
Generally, a word placed out of its normally expected position is somehow emphasised. This kind of thing is just the beginning of the many subtleties which must be understood in order to compose good Latin prose.
Poetry is a bit more complicated, in that the rules of prosody must be observed at the same time, though the same sort of rule applies. Look at this (an elegiac couplet from Ovid's Fasti):
Filia consuetis ut erat comitata puellis / Errabat nudo per sua prata pede
— and you will see what I mean.