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