How do T-cells determine which cells they've already inspected?
Your question spans two activities of T-cells that are related to each other: migration and activation. T-cells that usually stay in lymphoid organs migrate to non-lymphoid organs with different mechanisms for each T-cell subtype. When migrated to non-lymphoid organ, the T-cells move through the organ looking for infected cells.

Migration

As you can see in the image below, there are many mechanisms for moving T-cells across the endothelial layer.



The main idea for all three mechanisms is that the tethering of the T-cell to the endothelial layer is weak. Because of the weak interaction, the blood flow causes the T-cell to move across the endothelial layer while still being attached to the endothelial layer. This is known as rolling. Other stimuli (usually chemokines) such as CCL21, CCL25, or ICAM 1 are needed to induce T-cell to migrate across the endothelial layer. This is important because these chemokines are expressed where there is inflammation. For example, there is a correlation between CCL25 level in the gut and inflammation in the area.

T-cells express α4β7 integrin or CCR7 that bind T-cell to the endothelial layer. Central memory T-cells and naive T-cells express CCR7 and CD62L and thus reside preferentially within the secondary lymphoid organs. Effector memory T-cells bind to gut endothelium with α4β7 integrin, CCR9, and LFA-1.

CCL21 is expressed by both stromal cells of lymph node paracortex and endothelium of lymphatic vessels to assist migration of activated dendritic cells and naive and central memory T-cells to lymph node respectively. Dendritic cells are professional antigen-presenting cells (APCs) that migrate to lymph node where it can interact with T-cells more efficiently.

T-cell activation

One important concept to take away from T-cell activation is that there needs to be direct contact between MHC complex and T-cell receptor (TCR).

T-cells that have moved into the lymph node interact with dendritic cells, which have high MHC complex concentration on their cell surface. This leads to subsequent activation of the T-cell leading to release of cytokines such as IL-2 leading to proliferation of activated T-cells.

The activated T-cells now are able to freely move across the endothelial layer of non-lymphoid organs to infiltrate and look for infected sites. (Naive T-cells enter endothelial layer of non-lymphoid organs as well but this is chemokine independent nad thus not a reaction to inflammation).

Sources:

Difference between naive T-cells and memory T-cells: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1782715/

Dendritic cell's role in immune response: http://lab.rockefeller.edu/steinman/dendritic_intro/immuneResponse

Memory vs naive T-cell migration:
http://www.nature.com/icb/journal/v86/n3/full/7100132a.html

Dendritic cell and role of CCL21:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3078419/

Naive T-cell non-lymphoid organ infiltration: http://onlinelibrary.wiley.com/doi/10.1002/eji.200535539/full

CCL25 and inflammation relation:
http://www.sciencedirect.com/science/article/pii/S0896841116300014