Adding to what @Schmerel wrote, the Chanukah story probably took place during the time of Yose ben Yoezer & Yose ben Yochanan and possibly also during the time of their students, Yehoshua ben Perachya and Nitai Ha'arbeli.
There's a midrash in Beresheet Rabbah 65:22 which describes the death of Yose ben Yoezer:
" וְיָקוּם אִישׁ צְרוֹרוֹת הָיָה בֶּן אֲחוֹתוֹ שֶׁל רַבִּי יוֹסֵי בֶּן יוֹעֶזֶר אִישׁ צְרֵידָה, וַהֲוָה רָכֵיב סוּסְיָא בְּשַׁבְּתָא אֲזַל קוֹמֵי שָׁרִיתָא לְמִצְטַבָּלָא, אֲמַר לֵיהּ חֲמֵי סוּסִי דְּאַרְכְּבִי מָרִי וַחֲמֵי סוּסָךָ דְּאַרְכְּבֵךְ מָרָךְ. אָמַר לוֹ אִם כָּךְ לְמַכְעִיסָיו קַל וָחֹמֶר לְעוֹשֵׂי רְצוֹנוֹ, אָמַר לוֹ עָשָׂה אָדָם רְצוֹנוֹ יוֹתֵר מִמְּךָ, אָמַר לוֹ וְאִם כָּךְ לְעוֹשֵׂי רְצוֹנוֹ קַל וָחֹמֶר לְמַכְעִיסָיו. נִכְנַס בּוֹ הַדָּבָר כְּאֶרֶס שֶׁל עַכְנָא, הָלַךְ וְקִיֵּם בְּעַצְמוֹ אַרְבַּע מִיתוֹת בֵּית דִּין, סְקִילָה, שְׂרֵפָה, הֶרֶג וְחֶנֶק, מֶה עָשָׂה, הֵבִיא קוֹרָה נְעָצָהּ בָּאָרֶץ וְקָשַׁר בָּהּ נִימָא וְעָרַךְ הָעֵצִים וְהִקִּיפָן גָּדֵר שֶׁל אֲבָנִים, וְעָשָׂה מְדוּרָה לְפָנֶיהָ וְנָעַץ אֶת הַחֶרֶב בָּאֶמְצַע וְהִצִּית הָאוּר תַּחַת הָעֵצִים מִתַּחַת הָאֲבָנִים, וְנִתְלָה בַּקּוֹרָה וְנֶחְנַק, קִדְּמַתּוֹ הָאֵשׁ, נִפְסְקָה הַנִּימָה, נָפַל לָאֵשׁ, קִדְּמַתּוֹ חֶרֶב וְנָפַל עָלָיו גָּדֵר וְנִשְׂרַף. נִתְנַמְנֵם יוֹסֵי בֶּן יוֹעֶזֶר אִישׁ צְרֵידָה וְרָאָה מִטָּתוֹ פָּרְחָה בָּאֲוִיר, אָמַר בְּשָׁעָה קַלָּה קְדָמַנִּי זֶה לְגַן עֵדֶן."
Translation: And Yakum, man of Tz'rorot, was the nephew of Rabbi Yose ben Yoezer, man of Tzredah, and would ride a horse on Shabbat, and they brought before him a stake to hang upon it his uncle [Yose ben Yoezer]. Said he [Yakum]: "See the horse on which my master makes me ride, and see the horse (the gallows) on which thy master (the Lord) makes thee ride." [Yose] said to him: "If the men who angered Him receive such (pleasures in this world), then even more so to those who do His will (and will receive greater pleasures in the world to come)." [Yakum] said to him: "Is there a man that did His will more than you? (and yet you are about to suffer greatly. What pleasure is this?)" [Yose] said to him: "And if so receive those who followed His will, then even more so to those who angered Him!" And these words entered him like the venom of a snake, and so he went and fulfilled upon himself the four deaths of the court: stoning, burning, slaughter and hanging, what did he do? He brought a stake, stabbed it in the ground, tied to it a rope, prepared the sticks and surrounded them with a fence of stones, and he made a pyre before that fence and stuck the sword in the center and he lighted the fire beneath the sticks beneath the stones, and he hung himself from the stake and was choked, and the fire came upon him, and the rope was cut, and he fell to the fire, and the sword welcomed him, and the fence fell upon him and he was burned. As he was dying, Yose ben Yoezer closed his eyes and he saw his [Yakum's] bed floating in the air, [and] he said: "By [only] a short time did he precede me to Gan Eden."
This site presents all the sources related to the Chanukah story in chronological order and puts this story as happening around the time of Philipus's rule over Yerushalayim (in the site it's spelled Polyphus, but I think it's a mistake). Likewise, the story of the woman (later called Chanah) and her seven sons, according to that site, took place during this time. This was still before the actual Maccabean War. Jastrow also puts the midrash as happening during the time of Hellenist control (he thinks Yakum may have even been Alkimos (see also here), a central Hellenistic figure at the time) and the Maharzu writes on the midrash that he found in Seder Hadorot that Yose ben Yoezer was killed by the decree of Antiochus (possibly a reference to this, although elsewhere Seder Hadorot writes that he thinks Yose ben Yoezer lived a long life until he was killed in the time of King Yannai, but that Yochanan son of Matityahu was of the same generation as Yehoshua Ben Perachya and Nitai Ha'arbeli).
Rabbi Netanel Yosiphon wrote here that he thinks that after the death of Yose ben Yoezer, Chanukah was instituted either by his partner Yose ben Yochanan or by their students. Something of a drashic basis he finds in the teachings of Yose ben Yochanan and Yose ben Yoezer, because both spoke about the importance of the house:
"Yose ben Yoezer used to say: let thy house be a house of meeting for the Sages and sit in the very dust of their feet, and drink in their words with thirst. Yose ben Yochanan (a man) of Jerusalem used to say: Let thy house be wide open, and let the poor be members of thy household." (Avot 1:4-5)
And Chanukah has a certain focus on the "house": "נר איש וביתו", the Temple - the "house of Hashem", etc.
Rabbi Elisha Aviner also writes similarly that their focus on the house shows that they viewed the house as a spiritual center and base. Yose ben Yoezer thought that the way to fix the house would be to fill it with Torah, while Yose ben Yochanan thought the solution was to fill it with good middot.
Therefore, we may view their teachings as an attempt to rally the Jews to fight off the Greeks and the Hellenistic Jews, especially in order to save the Mikdash, but also in general, to save Am Yisrael from the negative spiritual influence of the Greeks. It's possible that their attempts to empower the Jews did have some influence in getting Jews to join the Chashmonaim.