I have an Alpine Series V. I am running P175/80R13, which is a 24.02" diameter and a 7" width. This size tire fits into the trunk spare well of an Alpine. I don’t know how a Tiger trunk is setup. Can’t answer the question what will fit.
From my research, the original 6.00r13 tires are 23.78" diameter and a width of 6.1". It is equivalent to P155/80R13. P205/60r13 is around 22.6" diameter. You are running at around 95% of the original diameter. 175/70R13 will have an equivalent diameter to your 205/60r13 tires. Tire Rack shows one 175/70r13 tire with an overall width of 6.2". It looks like the same width as the original tire.