A couple of sources prove that Aharon did have his own staff.
The Midrash in Bamidbar Rabbah 18:23 tries to trace the origins of Aharon's staff and notes a machlokes (argument) as to who it came from:
וּמַטֵּה אַהֲרֹן (במדבר יז, כא), יֵשׁ אוֹמְרִים הוּא הַמַּטֶּה שֶׁהָיָה בְּיַד יְהוּדָה, שֶׁנֶּאֱמַר (בראשית לח, יח): וּמַטְךָ אֲשֶׁר בְּיָדֶךָ. וְיֵשׁ אוֹמְרִים הוּא הַמַּטֶּה שֶׁהָיָה בְּיַד משֶׁה וּמֵעַצְמוֹ פָּרַח, שֶׁנֶּאֱמַר (במדבר יז, כג): וְהִנֵּה פָּרַח מַטֵּה אַהֲרֹן
And the staff of Aharon (Bamidbar 17:21). Some say that it is the staff that was in the hand of Yehudah, as it is stated (Genesis 38:18), "And the staff in your hand." And some say that it is the staff that was in the hand of Moshe. And it blossomed on its own, as it is stated (Numbers 17:23), "Behold the staff of Aharon sprouted."
So according to the first view, Aharon did in fact have his own staff which originally belonged to Yehudah.
However, perhaps more convincingly, another support for the fact that he had his own staff can be provided by the Zohar on the verse you note in Shemos 7:9. The Zohar 2:28a clearly delineates between Moshe and Aharon's sticks. It writes there as follows:
שמות ז׳:ט׳ וְאָמַרְתָּ אֶל אַהֲרֹן קַח אֶת מַטְּךָ. מַאי טַעֲמָא מַטֵּה אַהֲרֹן, וְלא מַטֵּה מֹשֶׁה. אֶלָּא, הַהוּא דְּמֹשֶׁה אִיהוּ קַדִּישָׁא יַתִּיר, דְּאִתְגְּלִיף בְּגִנְתָּא עִלָּאָה בִּשְׁמָא קַדִּישָׁא, וְלָא בָּעֵי קוּדְשָׁא בְּרִיךְ הוּא לְסָאֳבָא לֵיהּ בְּאִינּוּן חוּטְרִין דְּחַרְשַׁיָּא
"And you shall say to Aharon: Take your stick" (Shemos 7:9). What is the reason (he was told to take) the stick of Aharon and not the stick of Moshe? Rather it was because Moshe's (staff) was more holy since it was cut/forged for the sake of a higher purpose with a holy name, and Hashem did not want it to become defiled (through touching) the staffs of the Egyptian magicians.
So it would seem from the Zohar that Aharon very clearly had his own staff, and his stick was chosen above Moshe's to perform the sign in front of Pharaoh and his court, lest Moshe's staff, which was of a holier nature, would become defiled in some way.