Age-related macular degeneration (AMD) is an eye disease that leads to the deterioration of the central vision area of the eye and can gradually result in vision loss in elderly individuals. Early identification of this disease can significantly impact patient treatment outcomes. Furthermore, given the increasing elderly population globally, the importance of automated methods for rapidly monitoring at-risk individuals and accurately diagnosing AMD is growing daily. One standard method for diagnosing AMD is using optical coherence tomography (OCT) images as a non-invasive imaging technology. In recent years, numerous deep neural networks have been proposed for the classification of OCT images. Utilizing pre-trained neural networks can speed up model deployment in related tasks without compromising accuracy. However, most previous methods overlook the feasibility of leveraging pre-existing trained networks to search for an optimal architecture for AMD staging on a new target dataset. In this study, our objective was to achieve an optimal architecture in the efficiency-accuracy trade-off for classifying retinal OCT images. To this end, we employed pre-trained medical vision transformer (MedViT) models. MedViT combines convolutional and transformer neural networks, explicitly designed for medical image classification. Our approach involved pre-training two distinct MedViT models on a source dataset with labels identical to those in the target dataset. This pre-training was conducted in a supervised manner. Subsequently, we evaluated the performance of the pre-trained MedViT models for classifying retinal OCT images from the target Noor Eye Hospital (NEH) dataset into the normal, drusen, and choroidal neovascularization (CNV) classes in zero-shot settings and through five-fold cross-validation. Then, we proposed a stitching approach to search for an optimal model from two MedViT family models. The proposed stitching method is an efficient architecture search algorithm known as stitchable neural networks. Stitchable neural networks create a candidate model in search space for each pair of stitchable layers by inserting a linear layer between them. A pair of stitchable layers consists of layers, each selected from one input model. While stitchable neural networks had previously been tested on more extensive and general datasets, this study demonstrated that stitching networks could also be helpful in smaller medical datasets. The results of this approach indicate that when pre-trained models were available for OCT images from another dataset, it was possible to achieve a model in 100 epochs with an accuracy of over 94.9% in classifying images from the NEH dataset. The results of this study demonstrate the efficacy of stitchable neural networks as a fine-tuning method for OCT image classification. This approach not only leads to higher accuracy but also considers architecture optimization at a reasonable computational cost.