The character of Satan
evolved over time. In between the Tanakh and the NT there was the Apocrypha.
"The evolution of the theory of Satan keeps pace with the development of Jewish angelology and demonology. In Wisdom ii. 24 he is represented, with reference to Gen. iii., as the author of all evil, who brought death into the world; he is apparently mentioned also in Ecclus. (Sirach) xxi. 27, and the fact that his name does not occur in Daniel is doubtless due merely to chance. Satan was the seducer and the paramour of Eve, and was hurled from heaven together with other angels because of his iniquity (Slavonic Book of Enoch, xxix. 4 et seq.). Since that time he has been called "Satan," although previously he had been termed "Satanel" (ib. xxxi. 3 et seq.). The doctrine of the fall of Satan, as well as of the fall of the angels, is found also in Babylonia (Schrader, l.c. p. 464), and is mentioned several times in the New Testament. Satan rules over an entire host of angels (Martyrdom of Isaiah, ii. 2; Vita Adæ et Evæ, xvi.). Mastema, who induced God to test Abraham through the sacrifice of Isaac, is identical with Satan in both name and nature (Book of Jubilees, xvii. 18), and the Asmodeus of the Book of Tobit is likewise to be identified with him, especially in view of his licentiousness. As the lord of satans he not infrequently bears the special name Samael. It is difficult to identify Satan in any other passages of the Apocrypha, since the originals in which his name occurred have been lost, and the translations employ various equivalents. An "argumentum a silentio" can not, therefore, be adduced as proof that concepts of Satan were not wide-spread; but it must rather be assumed that reference to him and his realm is implied in the mention of evil spirits of every sort (comp. Demonology, and Kautzsch, "Apokryphen," Index)."
SATAN - JewishEncyclopedia.com
References to Satan in the NT greatly outnumber those in the Tanakh. They can be seen to draw on the Apocrypha. This is especially the case with Revelation. Notice in the above quote from the Jewish Encyclopedia that in the Book of Wisdom Satan is portrayed as the author of all evil and apparently is the serpent in Eden.
Wisdom 2
23 For God created man to be immortal, and made him to be an image of his own eternity.
24 Nevertheless through envy of the devil came death into the world: and they that do hold of his side do find it.
The Apocrypha: Wisdom: Wisdom Chapter 2
Also from the above quote
"it must rather be assumed that reference to him and his realm is implied in the mention of evil spirits of every sort"
The 'later additions' you mention are
Jewish additions.