I'm not sure if Hamlet's handful of lines after the Ghost leaves at the end of Act I, scene 5, actually are his second soliloquy.
Remember that a soliloquy is a speech given by one character to himself. In contrast, monologue is a speech given by one character to other characters. Therefore, in Act I, scene v, Hamlet's lines are absolutely a soliloquy as there is no one with him after the Ghost exits. Hamlet delivers his soliloquy, "what else? / And shall I couple hell?" after the Ghost leaves, "Adieu, adieu! Hamlet, remember me. (Exit)," and before Horatio and Marcellus enter. Consequently, Hamlet is alone during this speech, there are no other characters present; therefore it is a soliloquy and his second one.
Here are the other major soliloquy's of the character Hamlet:
O, that this too too solid flesh would melt (1.2.131-61). -- Hamlet expresses his grief over the events.
O, what a rogue and peasant slave am I! (2.2.555-612) -- Hamlet criticizes himself for lack of action
To be, or not to be: that is the question (3.1.64-8). -- Hamlet ponders suicide
Tis now the very witching time of night (3.2.380-391) -- Hamlet encourages himself to confront his mother
Now might I do it pat, now he is praying (3.3.77-100) -- Hamlet watches the praying Claudius and considers action
How all occasions do inform against me (4.4.35-69) -- Hamlet complains that fate seems against his plan for revenge