Abstract GIFT is a lightweight cipher proposed by Banik et al. at CHES’17, motivated by the design strategy of PRESENT. GIFT-64[2021] is a variant of GIFT proposed by Sun et al. at EUROCRYPT’22 to achieve better resistance against differential attack while maintaining a similar security level against linear attack. At EUROCRYPT’22, Dong et al. proposed a new rectangle framework considering the key guessing strategies for linear key-schedule ciphers, and established a uniform automatic search model for the whole rectangle attack. In this paper, we extend it to be applicable to bit-oriented ciphers, and construct an automatic search model involved in the distinguisher and key-recovery phase for GIFT. Moreover, we utilize the key relations of the linear key-schedule to the model, and find some new distinguishers both for GIFT-64 and GIFT-64[2021]. To evaluate the probability more accurately, we propose a method to calculate the probability of the 2-round middle part which connects the boomerang distinguisher for GIFT, and apply it with the SAT method to evaluate the probability of the whole distinguishers. As a result, we search out a new 20-round related-key boomerang distinguisher for GIFT-64, and achieve a 26-round attack with better time complexity than the best previous attack. For GIFT-64[2021], we find a 20-round boomerang distinguisher and give the first 26-round rectangle attack under related-key scenario.