Conical intersections (CIs) play a fundamental role in photoreactions. Although it is widely known that CIs are not isolated points but rather multidimensional seams, there is a dearth of techniques to explore and characterize these seams beyond the immediate vicinity of minimum energy points within the intersection space (minimum energy conical intersections or MECIs). Here, we develop a method that connects these MECIs by minimal energy paths within the space of geometries that maintain the electronic degeneracy (the "seam space") in order to obtain a more general picture of a CI seam. This method, the seam space nudged elastic band (SS-NEB) method, combines the nudged elastic band method with gradient projected MECI optimization. It provides a very efficient way of finding minimum energy seam paths in the conical intersection seam. The method is demonstrated by application to two molecules: ethylene and the green fluorescent protein (GFP) chromophore. The results show that previously known MECIs for these molecules are connected within a single seam, adding further support to previous conjectures that all MECIs are topologically connected in the seam space. Analysis of the nonadiabatic dynamics further suggests that a broad range of seam geometries, not only the vicinity of MECIs, is involved in the nonadiabatic transition events. The current method provides a tool to characterize CI seams in different environments and to explore the importance of the seam in the dynamics.