Mussaf and Shaharit are not me'akev each other. The proper order is to say it after shaharit based on Rambam (Tefillah 9:13-4) and Shulchan Aruch 286:1. This is primarily based on the principle of Tadir VeSheino Tadir-the more common takes precedence as said by Rabbenu Yonah 19a says that even though Mussaf could be said all day it should be said after Shacharit.
If time for Shaharit is missed and its after 6.5 hours into the day:
(Yalkut Yosef shabbat 286.14) If an individual did not recite the musaf prayer until the time when one may recite minha prayer, he must recite the minha prayer before the musaf prayer. The earliest time one can pray minha is 6.5 hours into the day called Minha Gedola (the Tur 233:1 sets this as the primary time.) According to Rambam (Tefillah 3:2-3) the preferred time is 9.5 hours into the day and be'diavad 6.5 hours. The Shulchan Aruch 233:1, rules like the Rambam and Mishna Brurah (ibid), concurs with the Rambam and S'A, adding that if there is a need one can pray the earlier time.
As a result, this translates in to the normal order of minha (depending on your particular custom and whether it is Shabbat, Yom tov, Rosh Hodesh, etc..) at minimal ashre then uva-letzion (if it wasn't said in morning order) then the amida of minha then amida of mussaf. This is not considered a make-up for having missed mussaf, only that mussaf can be prayed anytime within this time frame, which again follows the above principle of the more common takes precedence.
If missed Amida of Shaharit and its before 4 hours into the day:
YY 286.12 says that if one prayed amida of mussaf before amida of shacharit he has fulfilled his obligation but be'diavad, in other words you can't intend to pray amida of mussaf prior to amida of shacharit. The proper order would be to pray shaharit first before mussaf.
S'A 52.1 and Yalkut Yosef 1.65: List the minimum one must pray in order to pray amida of shaharit together is the full Shema Yisrael with the blessings and then proceed with the full amida of shaharit. If one calculates more time then he adds additional passage (ad loc.) from the pesuke dezimra and if one calculates more time he adds more passages (ad loc.) and so on. As a result, if the tzibur already finished the amida of shaharit and one came late and you can still catch up with the tzibur to say the mussaf amida together then you can follow S'A 52.1 order of one that came late and can still catch up to pray together with the tzibbur. Otherwise you have to pray the normal order of Shaharit before you can pray mussaf.
If missed Amida of Shaharit and after 4 hours into the day but before 6 hours into the day:
RAMBAM (tefillah 3.1) time for shaharit extends till 4th hour if it is missed one errors and prays after, he fulfills his obligation be'deavad. RAMBAM (tefillah 3.5) Proper time for Mussaf is after shaharit and before the 7th hour. The proper order here would be the same as above for praying shaharit first before mussaf and the items that you can skip to catch up.
RAMBAM (tefillah 3.9) If one hasn't prayed shaharit until 6th hour of the day, it is proper to read minha twice. And in the case of mussaf, one will follows the order of praying minha then mussaf and then shaharit (YY 286.14)
If one wants to pray mussaf before minha, See the following link (halachipedia) Tosfot (Brachot 28a s.v. Halacha) quotes the Ri who says that one only has to pray Mincha before Mussaf if one plans on praying both at this time however if one plans to say Mincha later one may pray Mussaf now. However, the anonymous first opinion of Tosfot argues. The Rashba sides with the Ri (and infers it from the language of the Gemara) and quotes the Teshuvat Goanim which also agrees. The Rabbenu Yonah (Brachot 19a s.v. VeYesh) comes up with this same idea on his own. The Tur 286:3 quotes it as a dispute but then writes that the Rosh would pray by himself and leave the minyan in order to pray mussaf before the time for Mincha comes. The Bet Yosef 286:4 quotes all the above rishonim who hold like the Ri. The S”A 286:4 writes the simple halacha that Mincha precedes mussaf and then quotes the opinion of the Ri (implying that the halacha doesn’t follow the Ri).
If praying alone
There is no general leniency that I am aware of that allows you to skip parts of the order of Shaharit before proceeding to the order of Mussaf. The order of prayer is setup the way it is for many reasons and one can't break them unless he really knows what he is doing. This would have to be answered by your Rav or at least a competent Rav (if you don't have a personal Rav to go to) and depending on your personal situation he will recommend the path to follow.
