All perfect praise be to Allah, The Lord of The Worlds. May His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Hajj-ul-Tamattu is one of the three types of Hajj. According to jurists, a person performing this type must perform Umrah during the Hajj season. It is stated: "This type of Hajj takes place by making Iharam (state of ritual consecration) for Umrah during the Hajj season from the Miqat (principal boundary at which a pilgrim intending to perform Hajj or Umrah must enter the state of Ihram) of his/her country or other, and finish it then initiate Hajj from Mecca or the Miqat from where he/she made Ihram for Umrah or from a closer Miqat." [Moghni Al-Mohtaj Ela Ma`rifat Ma`ani Al-Fath Al-Minhaj, Vol.2/P.287].
Usually, most pilgrims perform Hajj-ul-Tamattu by performing Umrah and Hajj while staying in Mecca. However, sometimes a Mutamatti might travel or go back to his/her country between Umrah and Hajj because there is a long time between them, for example. If he/she performed Umrah then returned to his/her country passing the Miqat then returned to perform Hajj, he/she remains as Mutamatti as long as he/she had made Ihram for Umrah during the Hajj season and performed Hajj in the same year and having travelled between the two rituals doesn`t make him/her other than Mutamatti. However, if he/she made Ihram for Hajj from the Miqat or before it then he/she is relieved from offering the animal sacrifice offered in Hajj-ul-Tamattu.
Al-Shirbini (May Allah be pleased with him) said: "The Mutamatti is liable for offering animal sacrifice because Allah Says (What means): "if any one wishes to continue the Umra on to the hajj, He must make an offering" [Al-Baqarah/196]. The obligation of offering an animal sacrifice is for having made one Ihram. Had he/she made Ihram for Hajj first from the Miqat of his/her country, he/she would have been obliged to go outside the boundaries of the Haram after finishing Hajj then make Ihram for Umrah. However, if he/she performed Hajj-ul-Tamattu, then he/she doesn`t have to go outside these boundaries because he/she makes Ihram for Hajj from any place inside Mecca." [Moghni Al-Mohtajj, Vol.2/P.288].
In conclusion, if the Mutamatti travelled to his/her country after having performed Umrah, then he/she remains Mutamatti. However, if he/she returned to perform Hajj and made Ihram from his/her country`s Miqat or other Miqat then he/she is relieved from offering the animal sacrifice. Nevertheless, if he passed the Miqat and made Ihram from Mecca then he/she is liable for offering an animal sacrifice for passing the Miqat without making Ihram, but he/she isn`t liable for offering animal sacrifice due in Hajj-ul-Tamattu. And Allah The Almighty Knows Best.