You assume correctly that it is fundamentally important to put on the tefillin shel yad before the shel rosh. The gemara[1] derives this from the p’sukim about the mitzva of tefillin, which all mention tying the tefillin to the arm before placing the tefillin on the head. In fact, despite the concept that one should not “pass over” one mitzva in order to do another first,[2] if one initially picked up the tefillin shel rosh, he should put it down in favor of the shel yad.[3] The gemara continues that we should take off the shel rosh before the shel yad, deriving this from the pasuk “v’hayu l’totafot bein einecha” (they shall be tefillin [shel rosh] between your eyes). The plural form indicates that when the tefillin are on your head, there should be two tefillin on you.
Is it an actual transgression to have the shel rosh on without the shel yad, or are the aforementioned sources simply instructions for the optimal order? There certainly cannot be an absolute transgression, as the mishna[4] states that in the absence of one of the tefillin, the other is put on by itself. This seems to indicate that the proper procedure for fulfilling the mitzva is to put on the shel yad before the shel rosh, but each individually still has value. This helps us understand the Taz’s[5] claim that if one mistakenly put on the shel rosh before the shel yad, then a greater concern than having the right order is to get to the state of having both on at the earliest opportunity. One accomplishes this not by first removing the shel rosh, which delays matters, but by putting on the shel yad immediately. The Taz’s opinion is accepted as halacha.[6]
Not all agree with this view, however. The Avudraham[7] compares the case of one who puts the shel rosh on before the shel yad to a situation in which a community reads the Chanuka Torah reading before that of Rosh Chodesh when the two coincide. He cites a source that indicates that one does not fulfill the mitzva of tefillin in the case of mistaken order, and he posits that the same is true for the Torah reading. In the context of Chanuka, the Rama[8] accepts the Avudraham’s opinion. The Taz takes issue with the Avudraham, however, firstly because the apparently Talmudic source that he cites is not found in our texts. It is also possible to accept the opinion of the Avudraham/Rama regarding Chanuka, based on local considerations, but not regarding tefillin.[9]
The Malbim,[10] in a minority opinion, accepts the Avudraham decision about tefillin. He cites statements of Chazal that indicate that putting on the shel rosh first is an infraction, and he maintains that if one did so, the situation should be rectified by removing the shel rosh. However, it appears that if one put on the tefillin in the right order and the shel yad later slipped, all would agree that one should return the shel yad to its proper position without removing the shel rosh. After all, at the time of his activity, the set of tefillin was put on in the correct order, and no infraction occurred.
The Shulchan Aruch[11] rules that one should make a beracha when returning tefillin that slipped out of place. However, many Acharonim do not accept this opinion. The Shelah explains that when making the original beracha, one’s intention is to have it extend until the end of tefilla, regardless of such breaks. Although there is no unanimity on the matter, that seems to be the more accepted practice.[12]
Therefore, you can simply return your tefillin shel yad to its proper place without any additional actions or repercussions.