I can only venture a partial answer to your question because, apart from considering whether this “trumpet call of God” is literal or symbolic, I cannot imagine what else it might be. My imagination does not stretch into realms beyond that of the literal or the symbolic. I also wish to preface my partial answer with a confession: I have never before made any study of this subject so the best I can offer is to provide quotations from an esteemed scholar who has. This is merely to kick-start your question, to get it going, so to speak, and I blush to do so as it’s really more of an up-start than a kick-start. However, here is the considered opinion of one, Matthew Henry, who did have interesting things to say about those three scriptures re. “the trumpet call of God”.
Matthew 24:31 – The angelic ministrations on that great day
“will be ushered in with a great sound of a trumpet, to awaken and
alarm a sleeping world. This trumpet is spoken of 1. Co. 15:52, and 1
Th. 4:16. At the giving of the law on mount Sinai, the sound of the
trumpet was remarkably terrible (Ex. 19:13, 16), but much more will it
be so in the great day. By the law trumpets were to be sounded for
the calling of assemblies (Num. 10:2), and in proclaiming the year of
jubilee, Lev. 25:9. Very fitly therefore shall there be the sound of a
trumpet at the last day, when the general assembly shall be called,
when the praises of God shall be gloriously celebrated, when sinners
shall fall as sacrifices to divine justice, and when the saints shall
enter upon their eternal jubilee.” [page 1386]
1 Corinthians 15:52 – “the trumpet must sound”
“It is the loud summons of all the living and all the dead, to come
and appear at the tribunal of Christ. At this summons the graves
shall open, the dead saints shall rise incorruptible, and the living
saints be changed to the same incorruptible state, v. 52.” [page 1822]
1 Thessalonians 4:16 – After detailing “the shout of a king” to mark his descent from heaven
“into this our air, v. 17. The appearance will be with pomp and
power, ‘with a shout’ – the shout of a king, and the power and
authority of a mighty king and conqueror, with ‘the voice of the
archangel’: an innumerable company of angels will attend him. Perhaps
one as general of those hosts of the Lord, will give notice of his
approach, and the glorious appearance of this great Redeemer and Judge
will be proclaimed and ushered in by the trump of God. For the trumpet
shall sound, and this will awaken those that sleep in the dust of the
earth, and will summon all the world to appear.” [page 1877]
Absolutely everything Henry writes speaks of literalism. He attaches no symbolic meaning to anything. Indeed, as there was a literal trumpet-blast at Sinai, and literal trumpet sounds for the calling of national assemblies and to signify the start of the year of jubilee, so should literal sounds be expected on that awesome Day. I personally doubt if the ancient shofar (ram's horn) or even silver trumpets (Numbers 10:10) would even approach the majesty and awe of angelic trumpets. Having been a horn and trumpet player long ago, I know something of the power of trumpets blown by humans, but nothing I know of will prepare me for the audio-impact of that great Day. I take this to be entirely literal just as there will be a literal descent from below the clouds of angels and the King of Glory, which every eye will see (Revelation 1:7) There are some who maintain that Jesus and his angelic millions will never be seen literally, who add that the trumpet sound will be equally symbolic, but such ones are on the fringes of orthodox Christianity. I recommend Matthew Henry’s “Commentary” on those verses.