One adds an alcohol like H-OCH3 to result in an ether, one adds a water to add an alcohol group.
For example, with double bond reactions, you can add H3O+ alone or you can add H2O using Hg(OAc)2, THF, followed up with NaBH4.
For adding an alkoxy group to a double bond, you don't usally add a protonated HOCH3 but instead add it with Hg(OAcF3), THF. I have never seen acid catalyzed alkoxylation to a double bond but I suppose it is possible.
The acid-catalyzed alkoxylation is an analogous reaction to the acid-catalyzed hydration (Markovnikov addition of water via acid catalysis), and can go as follows for a substituted alkene and a generic alcohol:
where the racemic mixture of the major or minor products can be written as a line bond instead of both the hash and wedge bonds.
(Had it not been racemic but uneven, a squiggly bond would have been the way to write it.)
The mechanism would go as follows:
with Markovnikov addition giving the major product.
Protonation of the alkene to create reaction conditions in which an alcohol can be a good nucleophile.
Nucleophilic backside-attack of the carbocation intermediate.
Removal of the attached alkoxide's proton (#"pKa" ~~ -3.6#) to regenerate the catalyst and form the product(s).