Some six letter words that rhyme with 'sigh' are:
undead
sticks
shinedinewhinelinepinevine
hobnob
patato
Six letter words starting with the letter "T"ThreatThreadTinkerTimingTakingTeamedToppedTriple
There are no English words ending in "-six", let alone six letter words.
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 3 words with the pattern RHYME-. That is, six letter words with 1st letter R and 2nd letter H and 3rd letter Y and 4th letter M and 5th letter E. In alphabetical order, they are: rhymed rhymer rhymes
The words six and lip do, but dice has a long I and a silent E, to rhyme with mice.
Six letter words starting with N:naggednailednapkinnationnaturenectarnearlyneaterneatlyneededneedleneckednegatenervesnestednestleneuternibblenicelynickelniecesnightsnimbleninetynogginnipplenobodynoodlenormalnoticenotionnovelsnovicenozzlenudgednuggetnursesnumbernurishnuzzlenylons
Six letter words that end with H:AmbushAttachAweighBanishBleachBlotchBluishBoyishBranchBreachBreathBreechBroachBroochBrunchCaliphChurchClinchClutchCrotchCrouchCrunchCrutchDanishDearthDetachDrenchFamishFetishFinishFlinchFourthFrenchGarishGlitchGrinchGrouchGrowthHaunchHealthHearthImpishKiboshKirschKitschLaunchLavishLengthLoofahLowishMenschModishMopishMulishNewishOnrushOwlishPariahParishPaunchPerishPlinthPloughPolishPreachPunishQuenchRadishRavishRelishRewashScorchScotchSearchSeraphSheathSketchSleighSleuthSlouchSnitchSpeechSplashSquashSquishStarchStenchStitchSwatchSwitchSwooshThatchThoughThrashThrushTrenchTroughTwitchVanishWarmthWealthWhooshWraithWreathWrenchZenith
Six (6) letter words that end with on:actionbaboonbeacon.beckon.bonbonbunionbuttoncannoncanyoncarboncartoncitroncocooncommoncottoncouponcrayondaikon.deacon.dragonepigon.falconflagonfusiongabion.gallongibbon.guidon.hagdon.jargonlagoonlegionlesionlessonlotionmaroonmatronmicronminionmotionmuttonnationneuronnotionoptionpardonparsonpatronpersonphotonpigeon.pistonpoison.potionprisonprotonpythonrationreasonreckonregionribbonsalmonsaloonseasonsermonsiphonsummontendontycoonunisonvisionwantonweapon.won tonzircon