Autism Spectrum Disorder (ASD) represents a lifelong developmental condition entailing significant healthcare expenses. Early identification of ASD traits has the potential to mitigate progression, but diagnosis methods, while reliable, are time-consuming and costly, and screening tools lack reliability. One potential approach to address challenges and expedite autism referrals and diagnosis is to leverage Artificial Intelligence (AI) algorithms and models. Autism AI is a project that aims to deliver high sensitivity to detect autistic children early. In the first phase of this project, we collected historical data from more than 11,000 participants answering multiple behavioral questions indicating autistic traits. However, only a small proportion of our participants indicated their formal ASD diagnosis, making our dataset largely unlabeled, which limits the utilization of supervised learning approaches. Here, we report on our preliminary study to design a hybrid semi-self-supervised training paradigm to design an ASD predictive Deep Neural Network using the Autism AI dataset. We initially trained the model via a small proportion of our data that contained diagnosis information and then proceeded with training with unlabeled data, using the pre-trained model to generate diagnosis predictions and optimize the network in a self-supervised manner. The initial results are promising, indicating noticeable sensitivity and specificity improvements over conventional and AI-informed screening techniques.