The nice thing about the Yamanote Line is that it runs - literally - every few minutes. Instead of the 8:10, you could get on the 8:07 and have a 13 minute connection, or the 8:05 and have 15 minutes, or the 8:02 and have 18 minutes - and so forth. You aren't limited to the exact train Hyperdia suggests. I rarely accept every link they recommend for the same reason - I want a bit more time for connections.

If this is a weekday, then please remember that Shinjuku and Shinagawa are likely to be mob scenes - especially Shinjuku. Do you HAVE to try this at 8:00 am? If so, I suggest an even earlier Yamanote train to make that 8:40 am Hikari.

Here is a trick you may not have figured out yet. On the Hyperdia screen that shows your connection - ie. "The First Course" or whichever - look across to the column named "Timetable" on the line for your Yamanote connection. See the icon of an open book with a clock? Click it and it will show EVERY train from your start (Shinjuku) to the intervening station (Shinagawa.) You can substitute any of the earlier trains to what Hyperdia suggested, giving yourself a longer connection. You can even look for alternatives to your Hikari suggestion by clicking the icon on that line.

If you are using a JR Rail Pass, you probably will want seat reservations for the shinkansen but you can just walk on whichever Yamanote train suits your needs. No ticket. If you are buying tickets, again - just reserve the shinkansen seat and use your Yamanote ticket for whichever train works for you.

JR