I was intrigued by this and began looking about. Mounce (MBG) counts 18 second perfects of one kind or another in the NT. Stevens writes on these bit also. Certainly the second perfect is not prevalent in the NT/LXX. It may be passing out of use by the time of κοινη but I'm not qualified to say one way or the other on that point. I am very interested to hear what answer you get from the tagging team Helen.
One last thought. I wonder if the second perfect is really going away or not. I have not studied this sufficiently, perhaps I can over time, but here is very interesting para or two from Robertson :
2. The Original Perfect. The Greek perfect is an inheritance from the Indo-Germanic original and in its oldest form had no reduplication, but merely a vowel-change in the singular.9 Indeed 191:6a (Sanskrit veda, Latin vidi, English wot) has never had reduplication.10 It illustrates also the ablaut from ιδ-, to οιδ-, in the singular, seen in Sanskrit and Gothic also.11 Cf. Latin capio, cepi (a to e). Note also κεῖ-μαι in the sense of τέ-θει-μαι. But the vowel-change characteristic of the original perfects is seen in other verbs which did use reduplication. Reduplication will receive separate treatment a little later, as it pertains to the present and aorist tenses also. It may be here remarked that the reduplicated form of some iterative presents doubtless had some influence in fastening reduplication upon the perfect tense. Note the English “mur-mur” (Greek γογ-γύζω, ἀρ-αρ-ίσκω), where the syllable is doubled in the repetition. It was a natural process. A number of these reduplicated forms with the mere change in the vowel appear in the N. T. This so-called second perfect, like the second aorist, is a misnomer and is the oldest form.1 In Homer indeed it is the usual form of the perfect.2 These old root-perfects, old inherited perfect forms according to Brugmann,3 persist in the κοινή and are reasonably common in the papyri,4 the inscriptions5 and the N. T. They are of two classes: (1) real μι perfects without any perfect suffix, like ἑστάναι (Ac. 12:14); (2) second perfects in — α, like γέγονα, λέλοιπα. As N. T. examples may be mentioned ἀκήκοα (Ac. 6:11), γέγονα (1 Cor. 13:1)), εἴωθα (Lu. 4:16), γέγραθα (Jo. 19:22), οἶδα (Jo. 10:4), ὄλωλα (ἀπ-,, Mt. 10:6), etc. These forms are found in the LXX. Cf. Helbing, Gr. d. Sept., p. 103; Thackeray, Gr., p. 252 f. But the κοινή gave up the shorter (without – α) forms of the plural indicative active perfect of ἵστημι (ἕσταμεν, ἕστατε, ἑστᾶσιν). See this chapter, iv, (d), 3, for details.
3. The κ Perfect. This is a new type created by the Greek language of which no adequate explanation has yet been offered. The Attic inscriptions already had the κ form (Meisterhans, p. 189 f.). It is apparently at first in the singular, as in ἕστηκα (pl. ἕσταμεν), etc.6 One might think that just as ἥκω has a perfect sense like κεῖμαι and finally had a few perfect forms7 (like ἥκασιν), so by analogy some κ verbs became the type and analogy did the rest. But Giles8 observes that the stems of the twelve or fourteen κ perfects in Homer all end in a vowel, a liquid or a nasal, not one in κ. And then the three κ aorists (ἔδωκα, ἔθηκα, ἧκα) call for explanation. But per contra there are some perfects in Homer which have κ stems like δέδορκα, ἔοικα, τέτηκα, etc. So that after all analogy may be the true explanation of the κ perfects which came, after Homer’s time, to be the dominant type in Greek. But the — κα perfects are rare in Homer. The examples are so common (δέδωκα, etc.), in the κοινή as in the classic Greek, as to need no list. Note ἕστηκα intransitive and ἕστακα transitive.
A. T. Robertson, A Grammar of the Greek New Testament in the Light of Historical Research (Accordance electronic ed. London: Hodder and Stoughton, 1919), 357-359.