I was at my yeshiva for Shabbat. The yeshiva's library has a number of sections that have interested me in particular in recent years - the "scholarly works" section, the Tanach section and the journals section. Whenever I stop by the yeshiva, I spend a few hours going through the same books and/or journals in these sections, because you never know what pearl might catch your eye this time.
So I found some potential leads:
1. In a book collecting Tanach essays by Rav Yaakov Medan, he brought in one of the essays the mishna in Pesachim 4:9
that criticized Chizkiyahu:
"The Sages taught: King Hezekiah performed six actions. With regard to three of them, the Sages of his generation conceded to him; and with regard to three of them, the Sages did not concede to him. Due to King Hezekiah’s father’s wickedness, he dragged the bones of his father Ahaz on a bier of ropes and did not afford him the respect due to a king, and the Sages conceded to him. He ground the copper snake that Moses fashioned in the desert because Israel worshipped it, and the Sages conceded to him. He suppressed the Book of Cures, and they conceded to him. And with regard to three actions, the Sages did not concede to him. He cut off the doors of the Sanctuary and sent them to the King of Assyria, and they did not concede to him because he thereby demeaned the Temple. He sealed the waters of the upper Gihon stream, diverting its water into the city by means of a tunnel, and they did not concede to him. He intercalated the year, delaying the advent of the month of Nisan during Nisan, and they did not concede to him."
He pointed out that the part "He sealed the waters of the upper Gihon stream, diverting its water into the city by means of a tunnel, and they did not concede to him" is very likely based on the following verses from Yeshayahu (22:11-12)
"and you constructed a basin between the two walls for the water of the old pool.
But you gave no thought to Him who planned it,
You took no note of Him who designed it long before.
My Lord GOD of Hosts summoned on that day
To weeping and lamenting,
To tonsuring and girding with sackcloth."
For me this was mind-blowing: Perhaps many such traditions about Tanach brought by the sages are simply pshat readings of Tanach!
Then it occurred to me: Wait, Why doesn't Rabbi Hillel bring any verses as evidence of his view?
I realized that maybe Rabbi Hillel did have a particular verse or verses in mind, but for some reason this was not quoted in the gemara.
So, one direction is to go through books that record the time of Chizkiyahu - Yeshahayhu, Micha, Melachim, Divrei Hayamim - and see if there's a verse that may be strongly connected to what Rabbi Hillel said.
2. Another book I found was one I had read in the past: Shki'in by Rabbi Shaul Lieberman. In the book, he analyzed Karaite and Christian sources and separated from them original ("Rabbinic") Jewish materials they quoted. I'm rereading now to see if he says anything regarding Rabbi Hillel. But he also has a very useful index of Jewish sources in Raymond Martini's Pugio Fidei, which is considered a very important anti-Jewish polemic because Martini spared no expense (well, he was state-church-funded...) to have dozens if not hundreds of Jewish sources translated into Latin. And yes, he does quote
at least once Rabbi Hillel. Martini identified him with Hillel, the brother of Rabbi Yehudah Nesi'ah.
What's interesting is that he brings a version that reads "בימי חזקיהו" with a Vav:
There are only two textual witnesses on the Genizah website that spell it this way - Vatican 171 and Florence 9-8, and Vat. 171 only spells it like that on 99a, not 98b.
Now, the above image is from an early printed edition, so I don't know whether the printer merely copied from the manuscript before him, or cross-referenced a Shas. And if a Shas - was it printed or a manuscript? It might be worth checking out the various Pugio MSS. to see how they brought that quote.
3. The above printed version of Pugio brought a book that's from after Martini's time, Rabbi Avraham Zaccuto's Sefer Yochasin, evidently to explain who were all of the sages mentioned by Martini. As it turns out, Rabbi Zaccuto thought that Rabbi Hillel was an amorah from the time of Rav Yosef.
Originally I thought that the full name was Rav Yosef bar Ha'madori, but it seems there's simply a missing period there. Otherwise we'd be faced with the mysterious question of: How did Rabbi Zaccuto know that Bar Ha'madori's full name was Rav Yosef bar Ha'madori (this is not stated anywhere in Shas) as well as how did he know that Rabbi Hillel was from his time/slash did he have a textual witness that read something like אמר רב יוסף בר המדורי שרא ליה מריה לר' הלל? But no, just a missing period...