Technology has transformed our physical interactions into infinitely more scalable and flexible digital ones. We can peruse an infinite number of photos, news articles, and books. However, these digital experiences lack the physical experience of paging through an album, reading a newspaper, or meandering through a bookshelf. Overlaying physical objects with digital content using augmented reality is a promising avenue towards bridging this gap. In this paper, we investigate the interaction design for such digital-overlaid physical objects and their varying levels of tangibility. We first conduct a user evaluation of a physical photo album that uses tangible interactions to support physical and digital operations. We further prototype multiple objects including bookshelves and newspapers and probe users on their usage, capabilities, and interactions. We then conduct a qualitative investigation of three interaction designs with varying tangibility that use three different input modalities. Finally, we discuss the insights from our investigations and recommend design guidelines.