Detailed magnetic field around a solenoid
The diagram below gives you a detailed look at the magnetic fields formed with loops that surround the wire and join to form larger loops that surround the coil.
The north and south mnemonic can be seen to clearly work . . .
. . . and so too the right hand grip rule can be seen to work clearly on each wire.
An easier way to see the effects of magnetic fields in a solenoid is to use one loop, as below.