Yes, and this is actually the most common approach. There are three types of حج, each handling the عمرہ-حج combination differently:
**حج Tamattu (Enjoying):** The most مستحب type for حجاج coming from outside مکہ مکرمہ. You enter احرام for عمرہ at the Miqat, perform عمرہ in مکہ مکرمہ (طواف, سعی, hair cutting), then exit احرام and enjoy normal life until the 8th of Dhul Hijjah, when you enter احرام again for حج from your accommodation in مکہ مکرمہ. This type requires an واجب animal sacrifice (hady). نبی کریم (صلی اللہ علیہ وسلم) said: 'Had I known what I know now, I would not have brought a sacrificial animal and would have made it an عمرہ' — encouraging Tamattu for those who did not bring animals.
**حج Qiran (Combining):** You enter احرام for both حج and عمرہ simultaneously at the Miqat. You remain in احرام from the time of عمرہ until after the حج rites are completed — a significantly longer period. The طواف and سعی of عمرہ are combined with those of حج. This type also requires an animal sacrifice.
**حج Ifrad (Single):** You enter احرام only for حج, without performing عمرہ. No animal sacrifice is required (though it is مستحب). You may perform عمرہ after completing all حج rites.
For most modern حجاج, Tamattu is ideal because it allows you to break from احرام between عمرہ and حج, providing comfort during the waiting days. The sacrifice is a small price for this significant convenience.